On k-folding map-germs and hidden symmetries of surfaces in the Euclidean 3-space
OOn k -folding map-germs and hidden symmetries ofsurfaces in the Euclidean 3-space Guillermo Peñafort Sanchis and Farid Tari
Abstract
Let M be a smooth surface in R (or a complex surface in C ) and k ≥ be an integer. At any point on M and for any plane in R , we construct aholomorphic map-germ ( C , → ( C , of the form F k ( x, y ) = ( x, y k , f ( x, y )) ,called a k -folding map-germ. We study in this paper the local singularities of k -folding map-germs and relate them to the extrinsic differential geometry of M .More precisely, we• stratify the jet space of k -folding map-germs so that the strata of codi-mension ≤ correspond to topologically equivalent A -finitely determinedgerms;• obtain the topological classification of k -folding map-germs on generic sur-faces in R (or C );• generalise the work of Bruce-Wilkinson on folding maps ( k = 2 );• recover, in a unified way, results obtained by considering the contact ofsurfaces with lines, planes and spheres;• discover new robust features on smooth surfaces in R . The aim of this work is to study k -folding map-germs on complex surfaces in C andrelate them to the extrinsic differential geometry of surfaces in R .The standard Whitney fold of order k with respect to the plane π : y = 0 in C isthe map ω k : C → C , given by ω k ( x, y, z ) = ( x, y k , z ) . a r X i v : . [ m a t h . DG ] F e b he map ω k ‘folds’ the space C along the plane π , gluing the points ( x, y, z ) , ( x, ξy, z ) , . . . , ( x, ξ k − y, z ) , where ξ = e πi/k is a primitive k th -root of unity. The Whit-ney fold of order k with respect to any plane π , denoted by ω πk , is defined similarly in§2. Let M be a complex surface in C . We call the restriction of ω πk to M the k -folding map on M with respect to π . As our study is local, given point p on M anda plane π in C through p , we choose a coordinate system so that M is locally thegraph of a function z = f ( x, y ) and π = π : y = 0 (see Remarks 2.3(4)). Then thegerm at p of the k -folding map is represented in standard form by the map-germ F k : ( C , → ( C , , given by F k ( x, y ) = ( x, y k , f ( x, y )) . (1)For an analytic (resp. smooth) surface M ⊂ R , the k -folding map at a point p on M is constructed by complexifying M (resp. a certain jet of a parametrisation of M )at p . The singularities of a k -folding map-germ encode the local symmetries of M withrespect to the (complex) reflection group of order k whose hyperplane arrangementconsists of the single plane π .The study of -folding map-germs on surfaces in R was carried out by Bruce andWilkinson [6, 10, 32] (see [8, 13, 14, 15, 16, 32] for more work on the subject), withoutresorting to complexification. The real map-germs in [6, 10, 32] are called foldingmap-germs and our -folding map-germs are their complexifications. Complexifyingdoes not give extra information when k = 2 . For k ≥ , per contra, k -folding mapsreveal a great deal of new geometric information. The local symmetries captured bythese map-germs cannot be seen in the real case, which is why we call them hiddensymmetries of M ⊂ R . The loci of their singularities are visible on M and captureextrinsic geometric information of the surface.Bruce and Wilkinson showed that folding maps capture the sub-parabolic and ridgecurves, as well as umbilic points and other special points on these curves: these arerobust features of the surface (i.e., they are special geometric features that can betraced on an evolving surface; see §6.2 for details). Passing to the complex setting, weshow that the singularities of k -folding maps, k ≥ , capture in a unified way, knownrobust features obtained by considering the contact of the surface with lines, planesand spheres (parabolic, sub-parabolic, ridge and flecnodal curves, umbilic points, B , C and S -points, A ∗ -points, cusps of Gauss (gulls-points) and butterfly-points). Ourapproach also reveals a new robust feature on surfaces: when k is divisible by ,we obtain a new curve, called the H -curve . We also obtain new special points onpreviously known curves as well as on the H -curve. This motivates the followingquestion: can the H -curve be obtained via the contact of the surface with somespecial geometric object? Further work is also required for understanding the linkbetween local (hidden) symmetries of a surface and its contact with lines and planes.2he paper is organised as follows. In §2, we set notation and give some prelimi-naries. In §3, we obtain formulae for the invariants C, T, µ ( D ) and r ( D ) of k -foldingmap-germs. These are respectively, the number of cross-caps, the number of triplepoints, the Milnor number and the number of branches of the double point curve.These invariants determine the finite A -determinacy and the topological class of a k -folding map-germ. In §4, we produce a stratification of the l -jet space of k -foldingmap-germs in standard form which is identified with the l -jet space of germs of func-tions J l (2 , . The stratification results are summarised as follows. Theorem 1.1
For any integer k ≥ , there is a stratification S k of J (2 , such that,for any stratum S in S k of codimension ≤ , all k -folding map-germs in standard formwith -jets in S are finitely A -determined and are pairwise topologically equivalent. We relate in §6 the stratification of the jet space to the extrinsic differential geom-etry of surfaces in R . After clarifying what it means for a surface to be generic, wededuce the following result about the topological classes of k -folding map-germs. Theorem 1.2
Let k ≥ be an integer and let M be a generic smooth surface in R ( or a complex surface in C ) . Then, at any point p on M and for any plane π through p , the k -folding map-germ at p with respect to π is finitely A -determined andis topologically equivalent to one of the following map-germs: M k ( x, y ) (cid:55)→ ( x, y k , y ) , M k ( x, y ) (cid:55)→ ( x, y k , xy + y ) , M kl ( x, y ) (cid:55)→ ( x, y k , y + y + x l y ) , l = 2 , , , N kl ( x, y ) (cid:55)→ ( x, y k , y + x y + y l − ) , l = 3 , , O k ( x, y ) (cid:55)→ ( x, y k , y + x y + xy ) , P kl ( x, y ) (cid:55)→ ( x, y k , xy + y + y l − ) , l = 2 , , , Q k ( x, y ) (cid:55)→ ( x, y k , xy + y + y + y ) , Q k ( x, y ) (cid:55)→ ( x, y k , xy + y + y + y ) , (cid:101) Q k ( x, y ) (cid:55)→ ( x, y k , xy + y + y + y ) , R k ( x, y ) (cid:55)→ ( x, y k , xy + y + y + y ) , U k ( x, y ) (cid:55)→ ( x, y k , x y + 2 xy + y + y ) , U k ( x, y ) (cid:55)→ ( x, y k , x y + 2 xy + y + y ) , V k,j,j (cid:48) ( x, y ) (cid:55)→ ( x, y k , x y + xy + a j,j (cid:48) y + b j,j (cid:48) x y + y ) , W k,j ( x, y ) (cid:55)→ ( x, y k , x y + xy + c j y + 4 x y + y ) , W p,p ( x, y ) (cid:55)→ ( x, y k , x y + y + y + y ) , X k ( x, y ) (cid:55)→ ( x, y k , xy + y + x y + y ) , Y k ( x, y ) (cid:55)→ ( x, y k , − xy + x y + y + y ) , where the constants a j,j (cid:48) , b j,j (cid:48) ( resp. c j ) are as in Proposition 4.23 ( resp. and j, j (cid:48) = 1 , . . . , k − , with j (cid:54) = j (cid:48) . k2 M k4 M k3 N k3 N k4 O k4 (B -point) P k2 M k2 P k3 Hyperbolic regionElliptic region P k4 Q k4 R k4(Butterfly-point) (S -point) (C -point) sub-parabolic curve ridge ridge flecnodalH -curve Q k4 ˜(H -point) Q k3 (A -point) N k4 M k4 M k2 U k3 U k4 X k4 Y k4(Cusp of Gauss)3p,p4 X k4 W k,j4 V k,j,j'4 M k3 N k3 U k3 curve Figure 1: Robust features captured by k -folding map-germs away from umbilic pointsfor k ≥ , even and divisible by (see Theorem 6.5 and Remarks 6.6 for the robustcurves at umbilic points).It is worth noting that -folding map-germs can have A -simple singularities andtheir corresponding strata in S k are A -constant. For k ≥ , none of the k -foldingmap-germs are A -simple, except for immersions and for the C -singularity of F , andthe strata of S k give rise to moduli of finitely determined map-germs with constantinvariants C, T, µ ( D ) and r ( D ) .The robust features captured by k -folding map-germs on a generic surface aresketched in Figure 1. An interesting finding is that, having studied symmetries ofinfinitely many orders (for any k ≥ ), we obtain a finite collection of robust featuresthat occur along curves and a finite collection of special points on these curves if wediscard the V k,j,j (cid:48) and W k,j -points. 4 Preliminaries
We introduce here k -folding map-germs and notation from singularity theory that areneeded in the paper. We start with the singularity theory notation, see for example[25, 31] for more details. We deal with germs F : ( M, p ) → ( N, F ( p )) of holomorphic maps between complexmanifolds. Taking coordinate chartes, this is the same as dealing with map-germs ( C n , → ( C p , .Let O n be the local ring of germs of holomorphic functions ( C n , → C and m n its maximal ideal (which is the subset of germs that vanish at the origin). Denote by O ( n, p ) the O n -module of holomorphic map-germs ( C n , → C p , so O ( n, p ) = (cid:76) p O n .Let R (resp. L ) be the group of bi-holomorphic germs ( C n , → ( C n , (resp. ( C p , → ( C p , ). The group A = R × L of right-left equivalence acts on m n . O ( n, p ) by ( h , h ) .G = h ◦ G ◦ h − . Two germs H, G are said to be A -equivalent, and write H ∼ A G , if H = ( h , h ) .G for some ( h , h ) ∈ A .The l -jet space of map-germs in m n · O ( n, p ) is by definition J l ( n, p ) = m n · O ( n, p ) / m l +1 n · O ( n, p ) . Given a germ G ∈ m n · O ( n, p ) , we identified its l -jet j l G with its Taylor polynomialof degree l at the origin. Let A l be the subgroup of A whose elements have l -jets thegerm of the identity. The group A l is a normal subgroup of A . Define A ( l ) = A / A l .The elements of A ( l ) are the l -jets of the elements of A . The action of A on m n . O ( n, p ) induces an action of the jet group A ( l ) on J l ( n, p ) as follows. For j l G ∈ J l ( n, p ) and j l ( h , h ) ∈ A ( l ) , j l ( h , h ) .j r G = j l (( h , h ) .G ) . A germ G is said to be finitely A -determined if there exist an integer l such that G ∼ A H for any H with j l H = j l G ; j l G is then said to be a sufficient jet of G . Thegerm G is then said to be l - A -determined. The least l satisfying this property is calledthe degree of determinacy of G .There are classifications of finitely determined map-germs for various pairs ( n, p ) .When p = 1 , there is Arnold’s extensive list of the R -classification of germs of func-tions ([1]). For ( n, p ) = (2 , , classifications were carried out by several authors,the most extensive ones are given in [17, 29]. Here we need only the singularities of A e -codimension ≤ , which we reproduce in Table 9. For ( n, p ) = (2 , , Mond [23]produced an extensive list of finitely A -determined map-germs. We use in this paperthe following singularities from [23]: 5mmersion ( x, y, Cross-cap ( x, y , xy ) S k ( x, y , y + x k +1 y ) , k ≥ B k ( x, y , x y + y k +1 ) , k ≥ C ( x, y , xy + x y ) H k ( x, xy + y k − , y ) , k ≥ X ( x, y , x y + xy + y ) The notion of a simple germ is defined in [1] as follows. Let X be a manifold and G a Lie group acting on X . The modality of a point g ∈ X under the action of G on X isthe least number m such that a sufficiently small neighbourhood of g may be coveredby a finite number of m -parameter families of orbits. The point g is said to be simple if its modality is , that is, a sufficiently small neighbourhood intersects only a finitenumber of orbits. The modality of a finitely A -determined map-germ is the modalityof a sufficient jet in the jet-space under the action of the jet-group.We also need the notion of topological equivalence. We say that two germs H, G ∈ m n · O ( n, p ) are topologically equivalent if H = h ◦ G ◦ h − for some germs of home-omorphisms h and h of, respectively, the source and target. k -folding maps In all this paper, we fix the inner product (cid:104) a, b (cid:105) = (cid:80) i a i b i in C .Let π be an element of the affine Grassmannian Graff(2 , of planes in C . A plane π has equation (cid:104) q, v (cid:105) = d , where v is a fixed non-zero vector orthogonal to π and d isa fixed scalar. However, any non-zero scalar multiple of ( d, v ) gives an equation of π ,so π is identified with the class ( d, v ) ∈ C P of ( d, v ) ∈ C .Let π : (cid:104) q, v (cid:105) = d be a plane in C . The orthogonal projection of a point p ∈ C to π along the vector v is the point q = p + λv ∈ π with λ = ( d − (cid:104) p, v (cid:105) ) / (cid:104) v, v (cid:105) .Consider the map ω πk : C → C given by ω πk ( p ) = q + λ k v = p + λv + λ k v. If we take ( d (cid:48) , v (cid:48) ) = ( αd, αv ) , α ∈ C \ , as another representative of π = ( d, v ) ,then p + λ (cid:48) v (cid:48) + λ (cid:48) k v (cid:48) = p + α ( d −(cid:104) p,v (cid:105) )) α (cid:104) v,v (cid:105) αv + α k ( d −(cid:104) p,v (cid:105) ) k ) α k (cid:104) v,v (cid:105) αv = p + λv + λ k α − k v = q + λ k α − k v. Clearly, the map ω πk depends on the points on the line ( αd, αv ) ∈ C and not merelyon the class of the line ( d, v ) ∈ C P . However, all these maps are L -equivalent: thebi-holomorphic map q − λv (cid:55)→ q − λα − kk v composed (on the left) with the map ω πk with π represented by ( d, v ) gives the map ω πk with π represented by ( αd, αv ) . Therefore,the L -class of ω πk depends only on π . 6 efinition 2.1 The
Whitney fold of order k ( k -fold for short ) with respect to a plane π ∈ Graff(2 , is the L -class of the map ω kπ . We still denote by ω πk any representativeof ω πk obtained by choosing a representative ( p, v ) of π = ( d, v ) ∈ C P . A k -fold may be viewed as generalisation of the Whitney fold ( x, y , z ) (cid:55)→ ( x, y , z ) .While the Whitney fold folds the space along the plane { y = 0 } and identifies thepoints ( x, y, z ) and ( x, − y, z ) , the Whitney fold of order k with respect to a plane π represented by ( d, v ) is a generically a k -to-one branched cover, ramified along π , andidentifies k -tuples of points q − λv, q − ξλv, . . . , q − ξ k − λv, where ξ = e πi/k is aprimitive k th -root of unity and q ∈ π .The map ω πk can also be viewed as the quotient map associated to the actionof the cyclic group Z /k Z , regarded as a complex reflection group whose hyperplanearrangement consist of the single plane π . We regard Z /k Z as the group generatedby the order k complex reflection q − λv (cid:55)→ q − ξλv . Observe that, even though theplane π does not determine ω πk uniquely (it depends on the choice of a representativeof π = ( d, v ) ∈ C P ), the action of Z /k Z on C is determined uniquely by π .Given any subset X ⊆ C , ω πk ( X ) encodes the order k reflectional symmetries of X with respect to π . See [18] for a recent work on singular maps related to reflectiongroups. Definition 2.2
Let M ⊂ C be a complex surface, p a point on M and k ≥ aninteger. Given π ∈ Graff(2 , , the k -folding map-germ on M at p with respect to π is the A -class of the restriction of ω πk to M at p . We denote any representative of theclass by F πk : ( M, p ) → ( C , ω πk ( p )) . Remarks 2.3
1. All the map-germs F πk with π represented by ( αd, αv ) , α ∈ C , are A -equivalent as the maps ω πk are L -equivalent. Thus, the A -class of F πk depends onlyon π and not on the choice of a representative ( p, v ) of π = ( d, v ) ∈ C P . In all thepaper, we work with a representative of the A -class of F πk .2. If p / ∈ π , then F πk is the germ of an immersion. Thus, to obtain any meaningfullocal geometric information about the surface M we should take the plane π passingthrough the point p ∈ M .3. The image of F πk is the image by ω πk of the germ ( M, p ) , so for p ∈ π ∩ M , F πk captures order k local symmetries of M with respect to π . The aim of this paper is tounderstand how these local symmetries are captured by the A -singularities of F πk .4. Let p ∈ π ∩ M and ( d, v ) a representative of π . If v / ∈ T p M , then F πk is a germof an immersion and is A -equivalent to ( x, y ) (cid:55)→ ( x, y k , y ) . Suppose that v ∈ T p M .We choose a coordinates system in C so that p is the origin, the z -axis is alonga normal vector to M at p , the y -axis along v and the x -axis orthogonal to theprevious two axes. Then we can take M locally at p as the graph z = f ( x, y ) of someholomorphic map f in a neighbourhood U of the origin. In this coordinate system, wehave π = π : y = 0 . Consequently, the k -folding map-germ on M at p is the germ7 k = F π k : ( C , → ( C , , given in standard form F k ( x, y ) = ( x, y k , f ( x, y )) . Inview of this, we shall always take a given k -folding map-germ in standard form (1).5. Definition 2.2 is adapted as follows for the real case. When M is an analyticsurface in R , denote by M C ,p its local complexification at p and by π C the complexi-fication of π . The k -folding map-germ on M at p with respect to π is then defined asthe A -class of the restriction of ω π C k to M C ,p at p . When M is a smooth surface, weconsider the k -folding map-germ of a given jet of (a parametrisation of) M at p . We recall the definitions of some key A -invariants of map-germs ( C , → ( C , .These are the Milnor number of the double point curve µ ( D ) , the number of cross-caps C and the number of triple points T . We give formulae for computing these invariantsfor k -folding map-germs, and use the invariants to study the finite A -determinacy andtopological equivalence of these germs. We start by recalling the definition of the double and triple point spaces of a corankone map-germ F : ( C n , → ( C n +1 , from [20]. Any such germ can be writtenin a suitable coordinate system in the form F ( x, y ) = ( x, f n ( x, y ) , f n +1 ( x, y )) , with x = ( x , . . . , x n − ) ∈ ( C n − , and y ∈ ( C , .Given h ∈ O n , the iterated divided differences of h are defined as h [ x, y, y (cid:48) ] = h ( x, y (cid:48) ) − h ( x, y ) y (cid:48) − y ∈ O n +1 ,h [ x, y, y (cid:48) , y (cid:48)(cid:48) ] = h [ x, y, y (cid:48)(cid:48) ] − h [ x, y, y (cid:48) ] y (cid:48)(cid:48) − y (cid:48) ∈ O n +2 . The multiple point ideals of a map-germ F as above are defined as I ( F ) = (cid:104) f n [ x, y, y (cid:48) ] , f n +1 [ x, y, y (cid:48) ] (cid:105) ⊆ O n +1 ,I ( F ) = (cid:104) f n [ x, y, y (cid:48) ] , f n [ x, y, y (cid:48) , y (cid:48)(cid:48) ] , f n +1 [ x, y, y (cid:48) ] , f n +1 [ x, y, y (cid:48) , y (cid:48)(cid:48) ] (cid:105) ⊆ O n +2 . The double and triple point spaces of F are, respectively, D ( F ) = V ( I ( F )) , D ( F ) = V ( I ( F )) . By counting variables and generators, it follows that D ( F ) (resp. D ( F ) ) is acomplete intersection whenever it has dimension n − (resp. n − ).The double point space D ( F ) , as a subset of ( C n − × C × C , , consists of points ( x, y, y (cid:48) ) such that either y (cid:54) = y (cid:48) and F ( x, y ) = F ( x, y (cid:48) ) or y (cid:48) = y and F is singular at ( x, y ) . 8o define the source double point space, we assume that F is finite. Then, theprojection π : D ( F ) → C n − × C given by ( x, y, y (cid:48) ) (cid:55)→ ( x, y ) is also finite. As aconsequence, the image of π can be given a complex structure as the -th Fitting ideal F ( π ∗ O D ( F ) ) of the push forward module π ∗ O D ( F ) (see [26] for details). The sourcedouble point space D ( F ) is defined as the projection π ( D ( F )) endowed with thiscomplex space structure, that is, D ( F ) = V ( F ( π ∗ O D ( F ) )) ⊆ C n . To compute the source double point space of a k -folding map-germ, we need thefollowing result. Lemma 3.1
Let Z be a germ of an n -dimensional Cohen-Macaulay space and let h , . . . , h r in O Z be regular elements. Write X j = V ( h j ) and X = V ( h . . . h r ) . Let φ : Z → ( C n , be a germ of a morphism of complex spaces such that the restrictions φ | X j : X j → ( C n , are finite. Then F (( φ | X ) ∗ O X ) = (cid:81) rj =1 F (( φ | X j ) ∗ O X j ) . Proof
It is enough to prove the statement for the case r = 2 . We can assume that X and X have no common irreducible component as topological spaces. Indeed,consider the two subspaces X = V ( h − t ) and X = V ( h ) of Z × ( C , and the map φ × Id : Z × ( C , → ( C n +1 , . The spaces X and X have no common irreduciblecomponent. Moreover, if the statement holds for φ × Id , then it holds for φ . This isa consequence of the fact that Fitting ideals commute with base change (see Lemma1.2 in [26]). Now consider the disjoint union X (cid:116) X ⊆ Z (cid:116) Z and the commutativediagram X (cid:116) X X ( C n , αψ φ | X The map α is generically a local isomorphism (that is, a local isomorphism on aZariski open and dense subset) because X and X are assumed to have no commoncomponent. Moreover, both X and X (cid:116) X are Cohen Macaulay spaces, which impliesthat the ideals F ( ψ ∗ O X (cid:116) X ) and F (( φ | X ) ∗ O X ) are principal. Since α is genericallya local isomorphism, they are necessarily equal. The statement then follows from theequalities F ( ψ ∗ O X (cid:116) X ) = F (( φ | X ) ∗ O X ⊕ ( φ | X ) ∗ O X )) = F (( φ | X ) ∗ O X ) · F (( φ | X ) ∗ O X ) . (cid:50) Theorem 3.2
For a k -folding map-germ F k ( x, y ) = ( x, y k , f ( x, y )) , the double pointspace D is the zero locus V ( λ ) , where λ = (cid:81) k − j =1 λ j and λ j = f ( x, y ) − f ( x, ξ j y )(1 − ξ j ) y , or ≤ j ≤ k − . We have thus a decomposition D = (cid:83) k − j =1 D j , with D j = V ( λ j ) . Proof
The double point space D ( F k ) is the intersection of the zero loci of thedivided differences (( y (cid:48) ) k − y k ) y (cid:48) − y and f ( x, y ) − f ( x, y (cid:48) ) y − y (cid:48) . Since (( y (cid:48) ) k − y k ) / ( y (cid:48) − y ) = (cid:81) k − j =1 ( y (cid:48) − ξ j y ) , we conclude that, as a set, the space D ( F k ) is the union of the spaces D j = (cid:26) ( x, y, ξ j y ) ∈ ( C n − × C × C , | f ( x, y ) − f ( x, ξ j y )(1 − ξ j ) y = 0 (cid:27) , for j = 1 , . . . , k − . Each of the sets D j projects to V ( λ j ) , which shows that D ( F k ) = V ( λ ) as sets.To show the equality as complex spaces, observe that the possible dimension of D ( F k ) is one or two. If D ( F k ) has dimension two, then some branch D j has dimensiontwo. Therefore, the corresponding function λ j is identically zero, which in turn implies λ = 0 . Since F k is finite by construction, the projection D ( F k ) is finite, hence theimage of D j is a germ of a two dimensional analytic closed subset of ( C , , so is equalto ( C , . This implies that D ( F k ) = ( C ,
0) = V (0) .Suppose now that D ( F k ) has dimension one. This implies that the functions (cid:81) k − j =1 ( y (cid:48) − ξ j y ) and f [ x, y, y (cid:48) ] form a regular sequence. Applying Lemma 3.1 with Z = V ( f [ x, y, y (cid:48) ]) gives D ( F k ) = V (cid:16)(cid:81) k − j =1 F (( π | D j ) ∗ O D j )) (cid:17) , where the D j , j = 1 , . . . , k − ,are given the natural complex space structure. Each of the morphisms π | D j : D j → C consists of forgetting the third coordinate of the tuple ( x, y, ξ j y ) , and this implies F (( π | D j ) ∗ O D j )) = (cid:104) λ j (cid:105) . (cid:50) Now we introduce some results to we use to check finite A -determinacy of a k -folding map-germ F k and topological triviality in families of such germs. The first ofthese results was proven in [20] for corank one map-germs, then extended to arbitrarycorank in [21]. Theorem 3.3
A finite map-germ F : ( C , → ( C , is finitely A -determined if,and only if, its double point curve D is reduced. The decomposition D = (cid:83) k − j =1 D j in Theorem 3.2 can be used to compute µ ( D ) ,making it easier to apply Theorem 3.3 (and Theorem 3.10 below). We denote by D j · D j (cid:48) the intersection multiplicity of two distinct branches of the double point curve.Clearly, D j · D j (cid:48) = D j (cid:48) · D j , and D j · D j (cid:48) = dim C O (cid:104) λ j , λ j (cid:48) (cid:105) . roposition 3.4 A k -folding map-germ F k is finitely A -determined if, and only if,the Milnor numbers µ ( D j ) , j = 1 , . . . , k − , and the intersection multiplicities D j · D j (cid:48) of all pairs D j and D (cid:48) j , with j (cid:48) (cid:54) = j , are finite. In that case, µ ( D ( F k )) = k − (cid:88) j =1 µ ( D j ) + 2 k − (cid:88) j,j (cid:48) =1 j By Theorem 3.3, F k is finitely A -determined if, and only if, µ ( D ( F k )) is finite,equivalently, D ( F k ) has an isolated singularity. This occurs if, and only if, every branch D j has an isolated singularity and no pair of branches D j and D j (cid:48) , with j (cid:54) = j (cid:48) , have acommon component. Using the formula µ = 2 δ − r + 1 for plane curves (see [22]) andthe property δ ( X ∪ Y ) = δ ( X ) + δ ( Y ) + X · Y , we get µ ( D ( F k )) = 2 δ ( D ( F k )) − r ( D ( F k )) + 1= k − (cid:88) j =1 (2 δ ( D j ) − r ( D j ) + 1) − k + 2 + 2 k − (cid:88) j,j (cid:48) =1 ,j 1. Suppose that D j is a germ of a regular curve parametrised by aregular map-germ α : ( C , → ( C , . Then, D j · D j (cid:48) = ord( h j (cid:48) ◦ α ) , which is thedegree of the first non zero term in the Taylor expansion of λ j (cid:48) ( α ( t )) .2. If both D j and D j (cid:48) are regular curves, we refer to D j · D j (cid:48) as the order ofcontact between D j and D j (cid:48) . We have D j · D j (cid:48) = 1 if, and only if, the two curvesintersect transversally. Suppose they are tangential and parametrised, respectively, by t (cid:55)→ ( t, γ j ( t )) and t (cid:55)→ ( t, γ j (cid:48) ( t )) . Then D j · D j (cid:48) = ord( γ j − γ j (cid:48) ) . 3. Let F k be a finitely A -determined k -folding map-germ. Then any pair ofbranches D j and D (cid:48) j , with j (cid:54) = j (cid:48) , cannot have any common irreducible component,otherwise D ( F k ) would fail to be reduced. Hence, we have r ( D ) = (cid:80) k − j =1 r ( D j ) . The defining functions λ j of the branches D j of the double point curve play a majorrole in our study of finite A -determinacy and topological equivalence of k -folding map-germs. We take F k ( x, y ) = ( x, y k , f ( x, y )) and write, for any given integer p ≥ , j p f ( x, y ) = p (cid:88) q =1 q (cid:88) s =0 a qs x q − s y s . (2)11hen, j p − λ j = p (cid:88) q =1 q (cid:88) s =1 ϑ sj a qs x q − s y s − , (3)with ϑ sj = 1 − ξ sj − ξ j = 1 + ξ j + · · · + ξ ( s − j . The constants ϑ sj play a significant role in determining the singularity type of thegerms λ j and in computing D j · D j (cid:48) . The following properties are needed in §4. Lemma 3.6 The numbers ϑ sj satisfy the following properties: (1) ϑ sj = 0 if, and only if, k | sj . (2) ϑ sj = 1 if, and only if, k | ( s − j . (3) If ϑ sj = ϑ sj (cid:48) , then ϑ sj is either or . Proof We observe that ϑ j = 0 , ϑ j = 1 and, for any integers m, n , we have ϑ mj = ϑ ( m + n ) j if, and only if, k | nj . For (1) we take m = 0 and n = s , and for (2) we take m = 1 and n = s − .For (3), we observe that the constants ϑ sj lie in the images of the curves γ n : S ⊂ C → C given by γ n ( z ) = (1 − z n +1 ) / (1 − z ) = 1 + z + . . . + z n . We show that the selfintersection points of the curves γ n are and (for n ≥ ).Write z = e iθ , with θ ∈ [0 , π ) . Then γ n ( θ ) = x + iy , with ( x, y ) ∈ R , gives − e i ( n +1) θ = ( x + iy )(1 − e iθ ) . Therefore, cos(( n + 1) θ ) = 1 − x + x cos( θ ) − y sin( θ ) , sin(( n + 1) θ ) = − y + y cos( θ ) + x sin( θ ) . Now the identity cos(( n + 1) θ ) + sin(( n + 1) θ ) = 1 gives (1 − cos( θ ))( x + y − x ) − y sin( θ ) = 0 . (4)Suppose that y = 0 . Then equation (4) becomes (1 − cos( θ )) x ( x − 1) = 0 , so x = 0 or or θ = 0 .When x = 0 , we have − e iθ (cid:54) = 0 , so − e i ( n +1) θ = 0 . That gives θ = πjn +1 , j =1 , . . . , n . Therefore, γ n passes n -times through the origin.When x = 1 , we get e inθ = 1 , so θ = πjn , j = 1 , . . . , n − . Therefore, γ n passes ( n − -times through the point .When θ = 0 , we have γ n (0) = n + 1 and the curve has no self-intersections at thatpoint. See Figure 2 for the cases n = 5 , , . Suppose now that y (cid:54) = 0 . Then equation (4) can be written as cot( θ/ 2) =( x + y − x ) /y. This shows that, for any ( x, y ) ∈ R with y (cid:54) = 0 , there is at mostone θ ∈ [0 , π ) satisfying γ n ( θ ) = x + iy . Therefore, the only self-intersection pointsof γ n are and . (cid:50) γ n for n = 5 , , (from left to right). Remark 3.7 As k cannot divide j , the condition k | sj in Lemma 3.6 (1) can also bewritten as d = gcd( k, s ) (cid:54) = 1 and j ∈ { kd , . . . , ( d − kd } . Of course, the same applies tothe condition k | ( s − j in Lemma 3.6 (2). The number of cross-caps and triple points are invariants of finitely A -determinedmap-germs F : ( C , → ( C , that can be described using stable deformations.A stable mapping U → ( C , with U an open neighbourhood of the origin in C exhibits only regular points, transverse double points along curves, cross-caps andisolated transverse triple points. Every stable deformation F t of F exhibits the samenumber C of cross-caps and T of triple points [24]. For a corank one map-germ, theseare given by the formulae C = dim C O J F , T = 16 dim C O I ( F ) , where J F is the ideal generated by the × minors of the differential matrix of F [24] (the formula for C holds without the corank one assumption).For j, j (cid:48) ∈ { , . . . , k − } , with j (cid:54) = j (cid:48) , we set λ j,j (cid:48) = λ j − λ j (cid:48) y and define T j,j (cid:48) = dim C O (cid:104) λ j , λ j,j (cid:48) (cid:105) . Observe that T j,j (cid:48) = T j (cid:48) ,j for all j (cid:54) = j (cid:48) . Proposition 3.8 The number of cross-caps and of triple points of a finitely A -determined k -folding map-germ F k are given by C = dim C O (cid:104) y k − , ∂f∂y ( x, y ) (cid:105) and T = 13 (cid:88) ≤ j 0) = ∂f δ ∂x ( x, 0) = 0 , a condition that canbe avoided by choosing a suitable deformation f t of f .We take now an unfolding as above. Since Z is reduced, D (( ˜ F k ) δ ) is isomorphicto the union of the D j,j (cid:48) (( ˜ F k ) δ ) as complex spaces. The equality we need to show fol-lows now from the constancy of the numbers involved under continuous deformations.Defining λ δj and λ δj,j (cid:48) in the obvious way, we obtain dim C O I ( F k ) = (cid:88) ( x,y,y (cid:48) ,y (cid:48)(cid:48) ) dim C O I (( ˜ F k ) δ ) ( x,y,y (cid:48) ,y (cid:48)(cid:48) ) = (cid:88) ( x,y ) (cid:88) j (cid:54) = j (cid:48) dim C O (cid:104) λ δj , λ δj,j (cid:48) (cid:105) = (cid:88) j (cid:54) = j (cid:48) dim C O (cid:104) λ j , λ j,j (cid:48) (cid:105) . Since T j,j (cid:48) = T j (cid:48) ,j , we add only the numbers T j,j (cid:48) , with j < j (cid:48) , and replace / by / . (cid:50) .3 Topological triviality We say that two subspaces S and S (cid:48) of a topological space X have the same topologicaltype if there is a homeomorphism X → X restricting to an homeomorphism S → S (cid:48) .Milnor showed that two isolated hypersurface singularities with same topological typehave the same Milnor number. In the case of our invariants, a similar result can beobtained using the results in [28] and [30] (see [4] for details). Proposition 3.9 Let F, G : ( C , → ( C , be finitely A -determined map-germs. If F and G are topologically equivalent, then µ ( D ( F )) = µ ( D ( G )) , r ( D ( F )) = r ( D ( G )) , C ( F ) = C ( G ) and T ( F ) = T ( G ) . The Milnor number of the double point curve is enough to determine the topologicaltriviality of families of finitely A -determined map-germs. Proposition 3.10 (Corollary 40 [3]) A family of finitely A -determined map-germs G t : ( C , → ( C , is topologically trivial if, and only if, µ ( D ( G t )) is constant alongthe parameter t . Using the upper semi-continuity of the numbers involved, Proposition 3.10 can becombined with Proposition 3.4 to yield the following result. Corollary 3.11 Let F tk = ( x, y k , f t ( x, y )) be a family of finitely A -determined k -folding map-germs. The following statements are equivalent: (1) The family F tk is topologically trivial. (2) The numbers µ ( D j ) and D j · D j (cid:48) are constant along the family F tk . (3) The numbers C, µ ( D j ) , D j · D j (cid:48) and T j,j (cid:48) are constant along the family F tk . In this section, we study the singularities of k -folding map-germs which we take instandard form F k ( x, y ) = ( x, y k , f ( x, y )) (see Remarks 2.3(4)). We identify the set ofsuch germs with the set O of germs, at the origin, of holomorphic functions f . Foreach k , we obtain a stratification S k of J (2 , (and hence of J l (2 , for l ≥ ).The stratification consists of the strata of codimension ≤ stated in Theorem 1.1together with the complement of their union (i.e., the union of strata of codimension ≥ ). Every stratum of codimension ≤ of S k consists of finitely A -determinedand pairwise topologically equivalent k -folding map-germs. The jet level l = 11 isdetermined by the conditions defining the strata of S k which involve the coefficientsof f in (2) up to degree (see Tables 1, 2, 4, 6).15s pointed out in the introduction, the case k = 2 was studied in [6, 10, 32]. Thestratification S can be recovered from the results in this paper. The different A -classesobtain in [6] correspond to different topological classes. This follows by analysing theinvariants C, T, µ ( D ) and r ( D ) . We shall suppose here that k ≥ and write the jetsof f as in (2).It is clear that F k is an immersion if and only if a (cid:54) = 0 . All immersions are A -finitely determined and pairwise topologically equivalent. Moreover, only immer-sions have D = ∅ , hence r ( D ) = 0 . We define a (cid:54) = 0 as the open stratum of S k (corresponding to the jets of all immersions) and choose M k : ( x, y ) (cid:55)→ ( x, y k , y ) as a normal form for topological equivalence of germs in this stratum.The strata corresponding to singular germs are organized into four branches ac-cording to the following result. Lemma 4.1 For k ≥ , every singular k -folding map-germ is A -equivalent to a k -folding map-germ whose -jet is equal to ( x, , xy + y ) , ( x, , y ) , ( x, , xy ) , or ( x, , . Proof As we are assuming F k to be singular at the origin, we have a = 0 . Then F k is A -equivalent to a germ whose -jet is ( x, , a xy + a y ) . Depending on thecoefficients a and a , the -jet can be taken to one of the following forms: ( x, , xy + y ) ⇐⇒ a = 0 , a a (cid:54) = 0 (Branch 1) ( x, , y ) ⇐⇒ a = a = 0 , a (cid:54) = 0 (Branch 2) ( x, , xy ) ⇐⇒ a = a = 0 , a (cid:54) = 0 (Branch 3) ( x, , ⇐⇒ a = a = a = 0 (Branch 4) (cid:50) We have the following about A -simplicity of germs of k -folding maps. Proposition 4.2 There are no A -simple k -folding map-germs for k ≥ . Proof It is enough to show that the orbit of a map-germ F k with a 2-jet ( x, , xy + y ) is not simple as the orbits of germs in the remaining branches in Lemma 4.1 areadjacent to it. For such a germ, we have j k F k ∼ A ( k ) ( x, ( y − x ) k , y ) . The result followsby Theorem 1:1 in [23] as there are no A -simple germs of the form ( x, y , f ( x, y )) with j f ≡ .When k = 4 and for F in Branch 1, we have j F ∼ A (4) ( x, xy + x y, y ) . This is a C -singularity and is A -simple [23]. For F in Branch 2, we have j F ∼ A (4) ( x, , y ) soit leads to non A -simple germs. By adjacency, the germs in Branches 3 and 4 also leadto non A -simple germs. Therefore, the C -singularity is the only A -simple singularityof -folding map-germs.The case k = 3 is treated in §4.1 where there are several A -simple singularities of -folding map-germs. (cid:50) S of codimension ≤ . Normal form Defining equations and open conditions CodimImmersion a (cid:54) = 0 S a = 0 , a a (cid:54) = 0 S a = a = 0 , a a (cid:54) = 0 S a = a = a = 0 , a a (cid:54) = 0 S a = a = a = a = 0 , a a (cid:54) = 0 H a = a = 0 , a (cid:54) = 0 , CndH (cid:54) = 0 H a = a = CndH = 0 , a (cid:54) = 0 , CndH (cid:54) = 0 H a = a = CndH = CndH = 0 , a (cid:54) = 0 , CndH (cid:54) = 0 X a = a = a = 0 , a a a (cid:54) = 0 U a = a = a = a = 0 , a a (cid:54) = 0 , CndU m (cid:54) = 0 X a = a = a = a = 0 , a a a (cid:54) = 0 W , a = a = a = a = 0 , a a (cid:54) = 0 , a a − a a (cid:54) = 0 CndH = a a − a a CndH = a a − ( a a + a a ) a + a ( a a + a a ) a − a a a .CndH = a , a − ( a a + a , a + a a ) a +( a a a + a a a + a a a + 2 a a a + a a ) a − ( a a + a a a + 2 a a a + a a a + 2 a a a + a a ) a a +(2 a a a + a a a + 3 a a a ) a a − a a a CndU m = a ( a a − a a ) + a a (see Table 6) Remarks 4.3 1. The degree of A -determinacy of a singular germ F k is greater orequal to k (the germ ( x, y ) (cid:55)→ ( x, , f ( x, y )) is not finitely A -determined for any f ).2. For germs in Branch 1 or Branch 2, we have F k ∼ A ( x, yp ( x, y ) , y ) for somegerm p . One can study the germ p , as was done in [23], instead of F k , but this blursthe order of the original k -folding map-germ. Also, the approach in [23] of reducingthe action of A on the set of -folding map-germs to the action of a subgroup of K on m does not extend to germs of k -folding maps for k ≥ . k = 3 When k = 3 we get several A -simple map-germs. For this reason, we treat this caseseparately. Theorem 4.4 The only A -simple singularities a -folding map-germ F can have arethose of type S l − , l ≥ , or H s , s ≥ . In the real case, the S l − -singularities are oftype S − l − . The strata of S of codimension ≤ are given in Table 1 . Proof We can write f ( x, y ) = f ( x, y ) + yf ( x, y ) + y f ( x, y ) , for some germs ofholomorphic functions f i , i = 0 , , . Then F ∼ A ( x, y , yf ( x, y ) + y f ( x, y )) . Suppose that a = 0 ( F is singular) and a (cid:54) = 0 . Then F ∼ A ( x, y , y ( g ( x ) + y h ( x, y )) + y ( a + k ( x, y ))) g ∈ m , h ∈ O and k ∈ m . We can makesuccessive changes of coordinates in the target so that j p F ∼ A ( p ) ( x, y , yL ( x ) + a y ) for any p ≥ . It is not difficult to show that F is finitely A -determined if, andonly if, ord ( L ) = ord ( g ) = ord ( f y ( x, is finite. Suppose that this is the case anddenote by l that order. Then j l L ( x ) = a l x l , a l (cid:54) = 0 , and the change of coordinates y (cid:55)→ y − ( a l / a ) x l in the source yields j l +1 F ∼ A (2 l +1) ( x, y − a l / a ) yx l , a y ) .This is an S l − -singularity, and since it is (2 l + 1) - A -determined, we have F ∼ A ( x, y − yx l , y ) . In the real case, this is an S − l − -singularity.The above calculations show, in particular, that we do not get the simple singular-ities B ± l , C ± l and F whose 2-jets are A (2) -equivalent to ( x, , y ) .Similar calculations show that when a = a = 0 and a (cid:54) = 0 , we get an H l -singularity when the singularity of F is finitely A -determined.The remaining cases are studied in the same way as in the case k ≥ (Table 6). Weget three strata with topological normal forms U , X and W , (the germs in thesestrata are not A -simple) together with the stratum represented by the topologicalnormal form U , which is topologically equivalent to Mond’s singularity X (see §2and [23]). (cid:50) Remark 4.5 The invariants associated to the simple singularities can be found in[24]; those associated to U , X , W , are given in Table 7. k ≥ We consider here the case when k ≥ , which we divide into the four branches accordingto the A (2) -orbits in Lemma 4.1. In all that follows, j ∈ { , . . . , k − } , and subindicesof singularities indicate the codimension of the stratum. a = 0 , a a (cid:54) = 0 Theorem 4.6 Any germ F k of a k -folding map satisfying a = 0 and a a (cid:54) = 0 isfinitely A -determined and is topologically equivalent to M k : ( x, y ) (cid:55)→ ( x, y k , xy + y ) . The invariants take the values µ ( D ) = ( k − , C = k − , T = 0 and r ( D ) = k − and the double point curve of F k is the union of k − regular curves intersectingtransversally. Proof The functions defining the branches D j of the double point curve (see Theo-rem 3.2) are given by λ j = a x + (1 + ξ j ) a y + O (2) , where O ( l ) denotes a remainderof order l . Clearly, all of the branches D j are regular curves. As the scalars ξ j are pairwise distinct, the space D = D ( F k ) = (cid:83) j D j consists of k − regular curvesintersecting transversally at the origin. 18e have µ ( D j ) = 0 and D j · D j (cid:48) = 1 , and from Proposition 3.4 we obtain µ ( D ) =( k − . By Theorem 3.3, any germ F k satisfying the conditions in the statement ofthe theorem is finitely A -determined. These germs form a stratum of codimension 1defined by { a = 0 , a a (cid:54) = 0 } . Since this stratum is path connected, we concludeby Theorem 3.10 that it consists of topologically equivalent germs. We choose for atopological model the germ M k given in the statement of the theorem.By Proposition 3.11, it is enough to compute C and T j,j (cid:48) for M k as these invariantsare constant along the stratum. We have C = dim C O / (cid:104) y k − , x + 2 y (cid:105) = k − and T j,j (cid:48) = dim C O / (cid:104) x + (1 + ξ j ) y, (cid:105) = 0 . (cid:50) Remark 4.7 The singularity M is the A -simple singularity C (see the proof ofProposition 4.2).The proofs for the cases in the remaining branches follow by similar arguments usedin the proof of Theorem 4.6 (except for the calculations of T ). To avoid repetition, wehighlight only key differences in each case. The notation for the conditions that definethe strata are those indicated in the tables. a = a = 0 , a (cid:54) = 0 Theorem 4.8 The strata of codimension ≤ of finitely A -determined k -folding map-germs in the branch a = a = 0 , a (cid:54) = 0 are those given in Table 2 . The invariantsassociated to the germs in each stratum are given in Table 3 . Proof The result follows from Propositions 4.9, 4.11 and 4.12. (cid:50) Table 2: Strata of codimension ≤ in Branch 2. Name Defining equations and open condition Codimtogether with a = a = 0 , a (cid:54) = 0 M k , (cid:45) k a (cid:54) = 0 M k , | k a a (cid:54) = 0 M k , (cid:45) k a = 0 , a (cid:54) = 0 M k , | k a = 0 , a a (cid:54) = 0 M k , (cid:45) k a = a = 0 , a (cid:54) = 0 M k , | k a = a = 0 , a a (cid:54) = 0 N k , | k a = 0 , a (cid:54) = 0 , CndN A (cid:54) = 0 N k , | k a = CndN A = 0 , a (cid:54) = 0 , CndN A (cid:54) = 0 O k , | k a = a = 0 , a a (cid:54) = 0 CndN A = a − a a CndN A = 8 a a − a a a + 2 a a a − a a For germs in this branch, the map-germ ( x, y ) (cid:55)→ ( x, f ( x, y )) is finite and genericallytwo-to-one. Therefore, T = 0 for any finitely A -determined map-germ in this branch.19able 3: Topological invariants of germs in the strata in Table 2. Name C T µ ( D ) r ( D ) M k , (cid:45) k k − k − k − − k + 2 k − M k , | k k − ( k − k − 2) + 3 − k k M k , (cid:45) k k − k − k − − k + 2 k − M k , | k k − ( k − k − 2) + 4 − k k − M k , (cid:45) k k − k − k − − k + 2 k − M k , | k k − ( k − k − 2) + 5 − k k N k , | k k − k − k − 2) + 3 k N k , | k k − k − k − 2) + 5 k O k , | k k − k − k − 2) + 4 k + 1 The germs of the functions defining the double point branch D j is given by λ j = ϑ j a y + a x + ϑ j a xy + ϑ j a y + O (3) . The branch D j is thus regular if, and only if, ϑ j (cid:54) = 0 . By Lemma 3.6, ϑ j = 0 when k is even and j = k/ . Proposition 4.9 Any k -folding map-germ F k satisfying a = a = . . . = a l = 0 , a a ( l +1)1 (cid:54) = 0 , for some l ≥ , and a (cid:54) = 0 when k = 2 p , is finitely A -determined andis topologically equivalent to M kl : ( x, y ) (cid:55)→ ( x, y k , y + y + x l y ) . The invariants C , T , µ ( D ) and r ( D ) are as in Table 2 . All the double pointbranches are regular curves except for the branch D p , when k = 2 p , which has an A l − -singularity. We have D j · D j (cid:48) = l , for all j (cid:54) = j (cid:48) . Proof Fix an index j and assume that (cid:45) k or that k = 2 p but j (cid:54) = p . Then D j isa regular curve and can be parametrised by t (cid:55)→ ( t, γ j ( t )) , with γ j ( t ) = − a l +1 , (1+ ξ j ) a t l + O ( l + 1) . Clearly, any two distinct branches have order of contact equal to l .Suppose now that k = 2 p . As ϑ p = 0 , the coefficients of x s y in λ p vanish for all s ≥ . Moreover, since ϑ p = 1 , the function λ p is of the form λ p = a l +1 , x l + a y + y h ( y ) . This implies that λ p is R -equivalent to y + x l if and only if a (cid:54) = 0 , in which caseit has an A l − -singularity. As for the contact between branches, ord ( λ p ( t, − a l +1 , (1+ ξ j ) a t l + O ( l + 1)) = l, hence D j · D p = l . This determines µ ( D ) , and hence the topologicaltriviality and the constancy of the invariants along the stratum. (cid:50) Remarks 4.10 1. Branch 1 can be considered as a particular case of the strata M kl in Proposition 4.9 (the condition a (cid:54) = 0 is not needed when l = 1 ).2. When (cid:45) k , the term y in M kl is irrelevant for topological equivalence. Weinclude it to represent both k even and k odd by the same map-germ. We do this forall subsequent topological normal forms. 20 roposition 4.11 Suppose that k = 2 p . Any k -folding map-germ F k satisfying a = a = a = 0 , a a (cid:54) = 0 and the additional conditions in (a) or (b) below is finitely A -determined and is topologically equivalent to one of the germs N kl : ( x, y ) (cid:55)→ ( x, y k , y + x y + y l − ) , l = 3 , . The invariants µ ( D ( F k )) , C, T, r ( D ( F k )) are as in Table 3. The branches D j areregular curves for all j (cid:54) = p and D j · D j (cid:48) = 2 for all j (cid:54) = j (cid:48) . (a) If CndN A (cid:54) = 0 , then the branch D p has an A -singularity and the map-germis topologically equivalent to N k . (b) If CndN A = 0 and CndN A (cid:54) = 0 , then D p has an A -singularity and themap-germ is topologically equivalent to N k . Proof We have λ p = a x + a x + a xy + a x + a x y + a y + O (5) . Ithas an A -singularity if, and only if, CndN A (cid:54) = 0 . When CndN A = 0 , we need toconsider the -jet of λ p . A calculation shows that λ p has an A -singularity if, and onlyif, CndN A (cid:54) = 0 . In both cases, we have ord ( λ p ( t, − a / ((1 + ξ j ) a ) t + O (3))) = 2 for all j (cid:54) = p . (cid:50) Proposition 4.12 Suppose that k = 2 p . Any k -folding map-germ F k satisfying a = a = a = a = 0 and a a a (cid:54) = 0 is finitely A -determined and is topologicallyequivalent to O k : ( x, y ) (cid:55)→ ( x, y k , y + x y + xy ) . The codimension of the stratum is and the invariants µ ( D ( F k )) , C, T, r ( D ( F k )) are as in Table 3. The branches D j , j (cid:54) = p , are regular curves and D p has a D -singularity. We have D j · D j (cid:48) = 2 for the distinct regular branches and D j · D p = 3 ,for j (cid:54) = p . Proof The result follows from the fact that λ p = a x + a xy + O (4) . (cid:50) a = a = 0 , a (cid:54) = 0 Theorem 4.13 The strata of codimension ≤ of finitely A -determined k -foldingmap-germs in the branch a = a = 0 , a (cid:54) = 0 are those given in Table 4 . Theinvariants of the germs in each stratum are given in Table 5 . Proof The result follows from Propositions 4.14, 4.16 and 4.18. (cid:50) For any finitely A -determined k -folding map-germ in this branch, we have C =dim C O / (cid:104) y k − , a x + O (2) (cid:105) = k − . ≤ in Branch 3. Name Defining equations and open conditions Codimtogether with a = a = 0 , a (cid:54) = 0 P k , (cid:45) k a (cid:54) = 0 P k , | k a (cid:54) = 0 , CndH (cid:54) = 0 P k , | k a (cid:54) = 0 , CndH = 0 , CndH (cid:54) = 0 P k , | k a (cid:54) = 0 , CndH = CndH = 0 , CndH (cid:54) = 0 Q k , (cid:45) k , (cid:45) k a = 0 , a (cid:54) = 0 Q k , | k , (cid:45) k a = 0 , a (cid:54) = 0 , CndH (cid:54) = 0 Q k , (cid:45) k , | k a = 0 , a (cid:54) = 0 , CndQm (cid:54) = 0 Q k , | k a = 0 , a (cid:54) = 0 , CndH (cid:54) = 0 , CndQm (cid:54) = 0 Q k , | k , (cid:45) k a = CndH = 0 , a (cid:54) = 0 , CndH (cid:54) = 0 Q k , | k a = CndH = 0 , CndH (cid:54) = 0 , CndQm (cid:54) = 0 (cid:101) Q k , (cid:45) k , | k a = CndQm = 0 , a (cid:54) = 0 , CndQm (cid:54) = 0 (cid:101) Q k , | k a = CndQm = 0 , CndH (cid:54) = 0 , CndQm (cid:54) = 0 R k , (cid:45) k , (cid:45) k a = a = 0 , a (cid:54) = 0 R k , | k , (cid:45) k a = a = 0 , a (cid:54) = 0 , CndQm (cid:54) = 0 R k , (cid:45) k , | k a = a = 0 , a (cid:54) = 0 , CndRm (cid:54) = 0 R k , | k a = a = 0 , a (cid:54) = 0 , CndQm (cid:54) = 0 , CndRm (cid:54) = 0 CndQm = a a − a a CndQm = a a − a a CndRm = a a − a a Table 5: Topological invariants of germs in the strata in Table 4. Name C T µ ( D ) r ( D ) P k , (cid:45) k k − ( k − k − (2 k − k − k − P k , | k k − ( k − k − (2 k − k − 2) + 4 k − P k , | k k − ( k − k − (2 k − k − 2) + 10 k − P k , | k k − ( k − k − (2 k − k − 2) + 16 k − Q k , (cid:45) k, (cid:45) k k − ( k − k − (3 k − k − k − Q k , | k, (cid:45) k k − ( k − k − (3 k − k − 2) + 2 k − Q k , (cid:45) k, | k k − ( k − k − (3 k − k − 2) + 12 k − Q k , | k k − ( k − k − (3 k − k − 2) + 14 k − Q k , | k , (cid:45) k k − ( k − k − (3 k − k − 2) + 8 k − Q k , | k k − ( k − k − (3 k − k − 2) + 20 k − (cid:101) Q k , (cid:45) k, | k k − ( k − k − (3 k − k − 2) + 18 k − (cid:101) Q k , | k k − ( k − k − (3 k − k − 2) + 20 k − R k , (cid:45) k, (cid:45) k k − ( k − k − (4 k − k − k − R k , | k, (cid:45) k k − ( k − k − (4 k − k − 2) + 6 k − R k , (cid:45) k, | k k − ( k − k − (4 k − k − 2) + 24 k − R k , | k k − ( k − k − (4 k − k − 2) + 30 k − t (cid:55)→ ( γ j ( t ) , t ) , with γ j ( t ) = − a ϑ j a t − a ( ϑ j ϑ j a a − ϑ j a a ) t − a ( ϑ j ϑ j a a − ϑ j ϑ j a a a + ϑ j a ( a a − a a ) + ϑ j a a ) t + O (5) . The strata are determined by the contact between the branches of the double pointcurve which depend on ϑ sj as well as on the coefficients a pq . We start with the case a (cid:54) = 0 , where the strata depend on the divisibility of k by . Proposition 4.14 Suppose that a = a = 0 and a a (cid:54) = 0 . Any k -foldingmap-germ F k in case (a) or satisfying the additional conditions in (b) is finitely A -determined and is topologically equivalent to one of the map-germs P kl : ( x, y ) (cid:55)→ ( x, y k , xy + y + y l − ) , for l = 2 , , . The invariants µ ( D ) , C, T, r ( D ) are as in Table 5. We have contact D j · D j (cid:48) = 2 ,except for D p · D p when k = 3 p which is given in (b) . (a) If (cid:45) k , then F k is topologically equivalent to P k . (b) If k = 3 p , then the strata are as follows: (b1) If CndH (cid:54) = 0 , then D p · D p = 4 and F k is topologically equivalent to P k . (b2) If CndH = 0 and CndH (cid:54) = 0 , then D p · D p = 7 and F k is topologicallyequivalent to P k . (b2) If CndH = CndH = 0 and CndH (cid:54) = 0 , then D p · D p = 10 and F k istopologically equivalent to P k . Proof If (cid:45) k , then by Lemma 3.6 we have ϑ j (cid:54) = 0 and ϑ j (cid:54) = ϑ j (cid:48) for all j (cid:54) = j (cid:48) .This implies D j · D j (cid:48) = 2 for all j (cid:54) = j (cid:48) .If k = 3 p , then by Lemma 3.6 the equality ϑ j = ϑ j (cid:48) holds only when { j, j (cid:48) } = { p, p } . Again, we obtain D j · D j (cid:48) = 2 for all j (cid:54) = j (cid:48) with { j, j (cid:48) } (cid:54) = { p, p } .We have ϑ p = ϑ p ) = 0 , ϑ p = ϑ p ) = 1 and ϑ p = ϑ p ) (Lemma 3.6). Using theparametrisations of D p and D p , we get D p · D p = 4 if, and only if, a a − a a (cid:54) = 0 ,equivalently, CndH (cid:54) = 0 .When CndH = 0 , the exceptional branches D p and D p are parametrised by x = − a a y + β s y + O (9) , s = 1 , , with β − β (cid:54) = 0 if, and only if, CndH (cid:54) = 0 . Then, D p · D p = 7 .When CndH = CndH = 0 , the exceptional branches are parametrised by x = − a a y + β s y + O (11) , s = 1 , , with β − β (cid:54) = 0 if, and only if, CndH (cid:54) = 0 . Then, D p · D p = 10 . 23he values of T can be computed using the models P kl . We have λ j = x + ϑ j y + ϑ (3 l − j y l − and λ j,j (cid:48) = ( ϑ j − ϑ j (cid:48) ) y + ( ϑ l − ,j − ϑ l − ,j (cid:48) ) y l − . By Lemma 3.6, T j,j (cid:48) =dim C O (cid:104) x,y (cid:105) = 1 when (cid:45) k or when k = 3 p but { j, j (cid:48) } (cid:54) = { p, p } . For k = 3 p , we showthat ϑ l − ,p (cid:54) = ϑ l − , p so T p, p = 3 l − . (cid:50) Remark 4.15 The singularity P : ( x, y ) (cid:55)→ ( x, y , xy + y + y ) is topologically equiv-alent to the singularity T : ( x, y ) (cid:55)→ ( x, y , xy + y ) in [23]. Observe that according toProposition 4.14, if k is not divisible by , then the y l − term can be removed fromthe expression of P kl , without changing the topological class of the germ. Proposition 4.16 Suppose that a = a = a = 0 and a a (cid:54) = 0 . Any k -foldingmap-germ F k in case (a) or satisfying the additional conditions in (b) , (c) or (d) isfinitely A -determined and is topologically equivalent to one of the following map-germs: Q k : ( x, y ) (cid:55)→ ( x, y k , xy + y + y + y ) Q k : ( x, y ) (cid:55)→ ( x, y k , xy + y + y + y ) (cid:101) Q k : ( x, y ) (cid:55)→ ( x, y k , xy + y + y + y ) The invariants are as in Table 5. We have D j · D j (cid:48) = 3 except when j and j (cid:48) arein the sets J or J (cid:48) below. (a) If (cid:45) k and (cid:45) k , then there are no exceptional branches and the germs in thisstratum are topologically equivalent to Q k . (b) If k = 3 p and (cid:45) k , then J = { p, p } . (b1) If CndH (cid:54) = 0 , then D p · D p = 4 and F k is topologically equivalent to Q k . (b2) If CndH = 0 and CndH (cid:54) = 0 , then D p · D p = 7 and F k is topologicallyequivalent to Q k . (c) If k = 4 p , (cid:45) k , then J = { p, p, p } . (c1) If CndQm (cid:54) = 0 , then D j · D j (cid:48) = 5 , for all j, j (cid:48) ∈ J, j (cid:54) = j (cid:48) , and F k istopologically equivalent to Q k . (c2) If CndQm = 0 and CndQm (cid:54) = 0 , then D j · D j (cid:48) = 6 , for all j, j (cid:48) ∈ J, j (cid:54) = j (cid:48) ,and F k is topologically equivalent to (cid:101) Q k . (d) If k = 12 p , the exceptional contact between double point branches occurs whenthe indices are in J = { p, p } or J (cid:48) = { p, p, p } . There are three strata: (d1) If CndH (cid:54) = 0 and CndQm (cid:54) = 0 , then D j · D j (cid:48) = 4 (resp. D j · D j (cid:48) = 5 ) forall distinct pairs with j, j (cid:48) in J (resp. J (cid:48) ), and F k is topologically equivalentto Q k . If CndH = 0 , CndH (cid:54) = 0 and CndQm (cid:54) = 0 , then D j · D j (cid:48) = 7 (resp. D j · D j (cid:48) = 5 ) for all distinct pairs with j, j (cid:48) in J (resp. J (cid:48) ), and F k istopologically equivalent to Q k . (d3) If CndQm = 0 , CndQm (cid:54) = 0 , and CndH (cid:54) = 0 , then D j · D j (cid:48) = 4 (resp. D j · D j (cid:48) = 6 ) for all distinct pairs with j, j (cid:48) in J (resp. J (cid:48) ), and F k istopologically equivalent to (cid:101) Q k . Proof The branches D j can be parametrised by t (cid:55)→ ( γ j ( t ) , t ) with γ j ( t ) = − a ϑ j a t − a ( ϑ j ϑ j a a − ϑ j a a ) t − a ( ϑ j ϑ j a a − ϑ j ϑ j a a a − ϑ j ϑ j a a a + ϑ j a a ) t + O (6) . (a) If (cid:45) k , (cid:45) k , then ϑ j (cid:54) = ϑ j (cid:48) for all j, j (cid:48) with j (cid:54) = j (cid:48) (Lemma 3.6). Therefore, D j · D j (cid:48) = 3 for all distinct pairs.(b) If k = 3 p and (cid:45) p , then ϑ j (cid:54) = ϑ j (cid:48) for all distinct pairs with j or j (cid:48) not in J = { p, p } . For such pairs, D j · D j (cid:48) = 3 .Using the fact that ϑ p = ϑ p ) = ϑ p = ϑ p ) = 0 , ϑ p = ϑ p ) = 1 and ϑ p = ϑ p (cid:54) = ϑ p ) = ϑ p ) , the parametrisation of the exceptional branch D p becomes γ p ( t ) = − a a t − ϑ p ( a a − a a ) a t − ϑ p a ( a a − a a ) a t + O (6) . A parametrisation of D p is obtained by replacing ϑ p by ϑ p ) in the expressionof γ p . Therefore, D p · D p = 4 if, and only if, a a − a a (cid:54) = 0 , i.e., CndH (cid:54) = 0 .When CndH = 0 , we have γ p = − a a t − a a − a a a + a a a t + ϑ p CndH a t + O (8) . For γ p , we replace ϑ p by ϑ p in γ p . It follows that D p · D p = 7 if, and only if, CndH (cid:54) = 0 .(c) If k = 4 p and (cid:45) p , we have D j · D j (cid:48) = 3 for j or j (cid:48) not in { p, p, p } .The parametrisation of D p becomes γ p ( t ) = a /a t + ϑ p ( a a − a a ) /a t + O (6) , and similarly for γ p and γ p replacing p by p and p respectively. Consequently,the branches D p , D p , D p have pairwise order of contact if, and only if, a a − a a (cid:54) = 0 , i.e., CndQm (cid:54) = 0 . When CndQm = 0 , γ p = − a /a t − ϑ p ϑ p ( a a − a a ) /a t + O (7) , withsimilar ajustements as above for γ p and γ p . Therefore, the three exceptional brancheshave pairwise order of contact if, and only if, a a − a a (cid:54) = 0 , i.e., CndQm (cid:54) = 0 .(d) This follows by Lemma 3.6 and (b) and (c) above.The contact between the branches determines µ ( D ) , the topological types and theirassociated strata. It remains to compute T for each normal form.25or Q k we have λ j = x + ϑ j y + ϑ j y + ϑ j y , and λ j,j (cid:48) = ( ϑ j − ϑ j (cid:48) ) y + ( ϑ j − ϑ j (cid:48) ) y + ( ϑ j − ϑ j (cid:48) ) y . Using the properties of ϑ sj in Lemma 3.6, we have T j,j (cid:48) = 2 unless k = 3 p and j, j (cid:48) ∈ { p, p } , or k = 4 p and j, j (cid:48) ∈ { p, p, p } . In the first case weget T j,j (cid:48) = 3 and in the second T j,j (cid:48) = 4 .The invariant for the germ Q k differs from Q k only when k is divisible by . For k = 3 p , we have T p, p = 6 and T j,j (cid:48) = 4 if k = 4 q and j, j (cid:48) ∈ { q, q, q } . All otherindices j and j (cid:48) give T j,j (cid:48) = 2 .Similarly, for the germ (cid:101) Q k and for k = 4 p , we get T j,j (cid:48) = 5 if j, j (cid:48) ∈ { p, p, p } . If k = 3 q , then T q, q = 3 . All other indices j and j (cid:48) give T j,j (cid:48) = 2 . (cid:50) Remark 4.17 When k is divisible by , the germs Q k and (cid:101) Q k have the same invari-ants C, T, µ ( D ) and r ( D ) but they are not topologically equivalent as their associatedsets of contacts between double points branches are distinct (see [33]). Proposition 4.18 Suppose that a = a = a = a = 0 and a a (cid:54) = 0 . Any k -folding map-germ F k in case (a) or satisfying the additional conditions in (b) or (c) is finitely A -determined and is topologically equivalent to R k : ( x, y ) (cid:55)→ ( x, y k , xy + y + y + y ) . The invariants associated to the germs in the stratum are as in Table 5 . We have D j · D j (cid:48) = 4 except for the exceptional pairs of branches below. (a) If (cid:45) k and (cid:45) k , there are no additional conditions and no exceptional branches. (b) If k = 4 p , (cid:45) k and CndQm (cid:54) = 0 , then D j · D j (cid:48) = 5 for all distinct pairs with j, j (cid:48) in J = { p, p, p } . (c) If k = 5 p , (cid:45) k and CndRm (cid:54) = 0 , D j · D j (cid:48) = 5 for all distinct pairs with j, j (cid:48) in J = { p, p, p, p } . (d) If k = 20 p , CndQm (cid:54) = 0 and CndRm (cid:54) = 0 , then D j · D j (cid:48) = 5 for all distinctpairs with j, j (cid:48) in J = { p, p, p } or in J (cid:48) = { p, p, p, p } . Proof The branches D j can be parametrised by t (cid:55)→ ( γ j ( t ) , t ) with γ j ( t ) = − a ϑ j a t + a ( ϑ j ϑ j a a − ϑ j a a ) t + a ( a ( ϑ j ϑ j a a − ϑ j a a ) − (( ϑ j ) a − ϑ j a a ) ϑ j a ) t + O (7) . (a) If (cid:45) k and (cid:45) k , then ϑ j (cid:54) = ϑ j (cid:48) for all j, j (cid:48) with j (cid:54) = j (cid:48) (by Lemma 3.6).Therefore, D j · D j (cid:48) = 4 for all distinct pairs.(b) If k = 4 p and (cid:45) k , then ϑ p = 1 , ϑ p = ϑ p , and we can write γ p ( t ) = − a /a t + ϑ p CndQm /a t + O (6) . We get similarly γ p ( t ) and γ p ( t ) by substi-tuting ϑ p by, respectively, ϑ p ) and ϑ p ) in γ p . The result follows as ϑ s (cid:54) = ϑ s (cid:48) for s, s (cid:48) ∈ { p, p, p } , s (cid:54) = s (cid:48) . 26c) If k = 5 p and (cid:45) k , then ϑ p = 0 , ϑ p = 1 , ϑ p = ϑ p , so γ p ( t ) = − a /a t + ϑ p CndRm /a t + O (7) . The expressions γ sp , s = 2 , , , are obtaining by substitut-ing ϑ p by ϑ sp ) in γ p .(d) The case k = 20 p follows in a similarly way to that of Proposition 4.16(d).The calculations for T are similar to those in the proof of Proposition 4.16. Weobtain T j,j (cid:48) = 4 if k = 4 p and j, j (cid:48) ∈ { p, p, p } , and T j,j (cid:48) = 5 if k = 5 q and j, j (cid:48) ∈{ q, q, q, q } . All remaining indices j and j (cid:48) give T j,j (cid:48) = 3 . (cid:50) Remark 4.19 The singularities Q and R have the same invariants, but the contactof their double point branches shows that they are not topologically equivalent. a = a = a = 0 Theorem 4.20 The strata of codimension ≤ of finitely A -determined k -foldingmap-germs in the branch a = a = a = 0 are those given in Table 6 . Theinvariants of the germs in each stratum are given in Table 7 . Proof The result follows from Propositions 4.22, 4.23, 4.25, 4.26, 4.27. (cid:50) The functions defining the double point branches D j have the following initialterms: λ j ( x, y ) = a x + a ϑ j xy + a ϑ j y + O (3) . Consequently, the branches D j are singular. A branch D j has an A -singularity unlessthe discriminant ∆ kj of the quadratic part of λ j vanishes. We have ∆ kj = ( a − a a ) ξ j + 2( a − a a ) ξ j + a − a a . (5)An A -singularity is a transverse intersection of two regular curves. Two branches D j and D j (cid:48) with an A -singularity may have one or both of their components beingtangential (that is, the tangent cones of the two branches have a non-trivial intersec-tion). Taking the resultant of j λ j and j λ j (cid:48) with respect to one of the variables, wefind that this happens if, and only if, a a = 0 or Ω kj,j (cid:48) = a a (1 + ξ j + ξ j (cid:48) ) − a ( ξ j + ξ j (cid:48) + ξ j + j (cid:48) ) = 0 . (6)We have ξ j + ξ j (cid:48) = 0 or ξ j + ξ j (cid:48) + ξ j + j (cid:48) = 0 if, and only if, k = 3 p and j, j (cid:48) ∈ { p, p } . Therefore, if (cid:45) k or if k = 3 p and j, j (cid:48) / ∈ { p, p } , V (Ω kj,j (cid:48) ) is acodimension 1 algebraic variety in J l (2 , , for l ≥ . For such pairs, we set α j,j (cid:48) = ξ j + ξ j (cid:48) + ξ j + j (cid:48) (1 + ξ j + ξ j (cid:48) ) and α = a a a . (7)We have the following properties of ∆ kj and Ω kj,j (cid:48) when a a (cid:54) = 0 ; the case a a = 0 is dealt with in Propositions 4.26 and 4.27.27able 6: strata of codimension ≤ in Branch 4. Name Defining equations and open conditions Codimtogether with a = a = a = 0 U k , (cid:45) k a a (cid:54) = 0 , ∆ kj (cid:54) = 0 , Ω j,k − j (cid:54) = 0 U k , | k a a (cid:54) = 0 , a (cid:54) = 0 , ∆ kj (cid:54) = 0 , Ω kj,j (cid:48) (cid:54) = 0 U k , | k a a (cid:54) = 0 , a = 0 , ∆ kj (cid:54) = 0 , Ω kj,j (cid:48) (cid:54) = 0 , CndU m (cid:54) = 0 V k,j,j (cid:48) , (*) a a (cid:54) = 0 , Ω kj,j (cid:48) = 0 , a (cid:54) = 0 , a (cid:54) = 0 , CndV m j,j (cid:48) (cid:54) = 0 W k,j , (cid:45) k , (**) a a (cid:54) = 0 , ∆ kj = 0 , CndW A (cid:54) = 0 W k,j , | k , (**), (***) a a (cid:54) = 0 , ∆ kj = 0 , CndW A (cid:54) = 0 , a (cid:54) = 0 W p,p , k = 3 p a a (cid:54) = 0 , a = 0 , a (cid:54) = 0 , Cnd (cid:102) W m (cid:54) = 0 X k , (cid:45) k , (cid:45) k a a (cid:54) = 0 , a = 0 , a (cid:54) = 0 X k , | k , (cid:45) k a a (cid:54) = 0 , a = 0 , a (cid:54) = 0 X k , (cid:45) k , | k a a (cid:54) = 0 , a = 0 , a (cid:54) = 0 , a (cid:54) = 0 X k , | k a a (cid:54) = 0 , a = 0 , a (cid:54) = 0 , a (cid:54) = 0 Y k , (cid:45) k a a (cid:54) = 0 , a = 0 , a (cid:54) = 0 Y k , | k a a (cid:54) = 0 , a = 0 , CndY A (cid:54) = 0 , CndY m j (cid:54) = 0 , a (cid:54) = 0 ∆ kj = ( a − a a ) ξ j + 2( a − a a ) ξ j + a − a a Ω kj,j (cid:48) = a a (1 + ξ j + ξ j (cid:48) ) − a ( ξ j + ξ j (cid:48) + ξ j + j (cid:48) ) CndU m = a ( a a − a a ) + a a CndV m j,j (cid:48) = a a β j,j (cid:48) + a a (2 a a − a a ) α j,j (cid:48) + a a , with α j,j (cid:48) = ξ j + ξ j (cid:48) + ξ j + j (cid:48) (1+ ξ j + ξ j (cid:48) ) , β j,j (cid:48) = ξ j +3 j (cid:48) + ξ j + j (cid:48) + ξ j (cid:48) +2 ξ j + j (cid:48) +2 ξ j +2 j (cid:48) + ξ j +2 ξ j + j (cid:48) + ξ j + ξ j (cid:48) (1+ ξ j + ξ j (cid:48) ) .CndW A = ( a a + a a ) a − a ( a a + 2 a a ) a + 8 a a a Cnd (cid:102) W m = a a − a a CndY A = a − a a CndY m j = ( a a + a a − a a a ) ξ j + a (2 a a − a a ) ξ j + a a ( ∗ ) j < j (cid:48) ; ( j, j (cid:48) ) (cid:54) = ( p, p ) when k = 3 p ( ∗∗ ) If | k then j (cid:54) = k/ ∗ ∗ ∗ ) If k = 3 p , then j (cid:54) = p, p Name C T µ ( D ) r ( D ) U k , (cid:45) k k − ( k − k − k − k − 2) + 1 2 k − U k , | k k − ( k − k − k − k − 2) + 3 2 k − U k , | k k − ( k − k − k − k − 2) + 9 2 k − V k,j,j (cid:48) , (cid:45) k k − ( k − k − k − k − 2) + 7 2 k − V k,j,j (cid:48) , | k k − ( k − k − k − k − 2) + 9 2 k − W k,j , (cid:45) k k − ( k − k − k − k − 2) + 3 2 k − W k,j , | k k − ( k − k − k − k − 2) + 5 2 k − W p,p , k = 3 p k − ( k − k − k − k − 2) + 11 2 k − X k , (cid:45) k, (cid:45) k k − ( k − k − k − k − 2) + 1 2 k − X k , | k, (cid:45) k k − ( k − k − k − k − 2) + 2 2 k − X k , (cid:45) k, | k k − ( k − k − k − k − 2) + 3 2 k − X k , | k k − ( k − k − k − k − 2) + 4 2 k − Y k , (cid:45) k k − ( k − k − k − k − 2) + 1 2 k − Y k , | k k − k ( k − k − k − 2) + 3( k − 1) 2 k − Proposition 4.21 Suppose that a a (cid:54) = 0 and that (cid:45) k or k = 3 p and j, j (cid:48) / ∈{ p, p } . Then: (1) ∆ kj = ξ j ∆ kk − j , and if ∆ kj = 0 then ∆ ks (cid:54) = 0 for s / ∈ { j, k − j } , so the solutionsof ∆ kj = 0 in the k th -roots of unity come in pairs. (2) For ∆ kj to vanish requires α in (7) to belong to the real semi-line ( −∞ , . (3) α j,j (cid:48) = α j (cid:48) ,j for all pairs ( j, j (cid:48) ) . We have α j,j (cid:48) = α k − j,j (cid:48) − j = α k − j (cid:48) ,j − j (cid:48) and thepairs ( j, j (cid:48) ) , ( k − j, j (cid:48) − j ) , ( k − j (cid:48) , j − j (cid:48) ) are pairwise distinct. Furthermore, α kl,q = α kj,j (cid:48) if, and only if, ( l, q ) or ( q, l ) is one of those pairs. (4) α j,j (cid:48) is real if, and only if, j (cid:48) = k − j or j (cid:48) = 2 j . In that case, by (3) , α k − j, j isalso real. Then, α j,k − j = α j, j = α k − j, j = (1 + ξ j + ξ k − j ) − . (5) If α is real then Ω kj,j (cid:48) = 0 if, and only if, j (cid:48) = k − j or j (cid:48) = 2 j . Then by (3) , wealso have Ω kk − j, j = 0 . (6) If α is real, then Ω kj ,k − j = 0 implies ∆ kj (cid:54) = 0 for all j . Conversely, if ∆ kj = 0 ,then Ω kj,k − j (cid:54) = 0 for all j . (7) If ∆ kj ∆ kj (cid:48) (cid:54) = 0 and Ω kj,j (cid:48) (cid:54) = 0 for all j, j (cid:48) with j (cid:48) (cid:54) = j , then D j · D j (cid:48) = 4 . Proof (1) As ξ − j = ξ k − j , factoring out ξ j in (5) gives ∆ kj = ξ j ∆ kk − j .If a − a a = 0 , then for ∆ kj to vanish requires a − a a = 0 . This wouldimply a a = 0 . Therefore, under the hypothesis of the proposition, we can assumethat a − a a (cid:54) = 0 . Then ∆ kj = 0 ⇐⇒ ξ j + 2( α − α − ξ j + 1 = 0 α as in (7). If ξ j is a solution of the above quadratic equation, then so is ξ k − j = ξ − j . Therefore, for α fixed, if ∆ kj = 0 , then ∆ ks (cid:54) = 0 for s / ∈ { j, k − j } .(2) When a a (cid:54) = 0 , we can write ∆ kj = a a ((1 + ξ j ) α − ξ j + ξ j )) . Clearly, ∆ kj (cid:54) = 0 when ξ j = − . Thus, ∆ kj = 0 if, and only if, α = 4(1 + ξ j + ξ j ) / (1 + ξ j ) ,which shows that if ∆ kj = 0 then α must be real. The discriminant of the quadraticequation ( ξ j + 1) α − ξ j + ξ j + 1) = 0 in ξ j is α − , so α < as the solutions ξ j are not real.(3) Clearly as α j,j (cid:48) = ξ j + ξ j (cid:48) + ξ j + j (cid:48) (1+ ξ j + ξ j (cid:48) ) , we have α j,j (cid:48) = α j (cid:48) ,j . Factoring our ξ j (resp. ξ j (cid:48) ) from the numerator and denominator gives α k − j,j (cid:48) − j = α k − j,j (cid:48) − j = α k − j (cid:48) ,j − j (cid:48) .We now seek pairs ( l, q ) for which α l,q = α j,j (cid:48) . We know from the above that wehave at least 3 such pairs. To show that these are the only ones, we write α j,j (cid:48) = c + id (so ( c, d ) (cid:54) = (0 , ) and represent points on the unit circle, with − removed, in theform z = − t t + i t t and w = − s s + i s s , with t, s ∈ R . We set α z,w = z + w + zw (1 + z + w ) . The real and imaginary parts of α z,w − ( c + id ) vanish if, and only if, P ( c,d ) ( s, t ) = Q ( c,d ) ( s, t ) = 0 , where P ( c,d ) ( s, t ) = ( cs + 4 ds + s − cs − ds + 2 s + c + 1) t +4( s + 1)( ds − cs − d + s ) t − cs − s + 10 cs + 8 ds − c + 1) t − s + 1)( ds + 2 cs + 3 d − s ) t + cs − ds + s + 2 cs − ds − s + 9 c − ,Q ( c,d ) ( s, t ) = − ( ds − cs − ds + 4 cs + d ) t + 4( s + 1)( cs + 2 ds − c + 1) t +2(3 ds + 10 ds + 2 s − cs − d + 2 s ) t − s + 1)( cs − ds + 3 c − t − ds − cs − ds + 4 s − cs − d + 4 s. Observe that P ( c,d ) and Q ( c,d ) are symmetric polynomials. Their resultant withrespect to t vanishes if, and only if, s = 3 or R ( c,d ) ( s ) = 0 , with R ( c,d ) ( s ) = ( c + d − s − (9 c + 18 c d + 9 d + 2 c + 2 d + 8 c − s +(27 c + 54 c d + 27 d + 18 c + 18 d − c + 3) s − c − c d − d + 18 c + 18 d − c + 1 . We have s = 3 if, and only if, k = 3 p and w = ξ p or w = ξ p , and this is excludedfrom the hypotheses. The component R ( c,d ) of the resultant is a cubic polynomial in s provided c + d − (cid:54) = 0 , i.e., | α j,j (cid:48) | (cid:54) = 1 . Suppose that | α j,j (cid:48) | (cid:54) = 1 . Then thediscriminant of R ( c,d ) vanishes if, and only if, d = 0 (i.e., α j,j (cid:48) is real and this is treatedin (4) below) or δ R = 27 c + 54 c d + 27 d − c − d + 8 c − . For ( c, d ) in the interior region bounded by the curve δ R = 0 (see Figure 3), R ( c,d ) hasa unique solution in s . As we know that there are at least three distinct solution pairs30igure 3: The red curve is the discriminant of R ( c,d ) , the blue curve is the unit circleand the green line are the values of α p,w and α p,w when k = 3 p .to the problem, it follows that ( c, d ) must be in the exterior region ( R + ) bounded bythe curve δ R = 0 .Observe that s = 0 is a root of R ( c,d ) if, and only if, δ R = 0 . Therefore, the rootsof R ( c,d ) in s do not change sign in ( R + ) . Choosing any point in that region, we findthat they are all positive. It follows that R ( c,d ) has six roots ± s , ± s , ± s . Thesecorrespond to six points on the unit circle w , w , w and w , w , w . For each root of R ( c,d ) we show, by considering the subresultant (see for example[19]) of P ( c,d ) and Q ( c,d ) that P ( c,d ) ( ± s i , t ) and Q ( c,d ) ( ± s i , t ) have only one common root.As P ( c,d ) and Q ( c,d ) are symmetric polynomials, that common root is a root of R ( c,d ) .Interchanging w i with w i if necessary, we can set w = ξ j , w = ξ k − j , w = ξ k − j (cid:48) , Thenthe solutions of α z,w − c − id = 0 are exactly, up to permutation of z and w , ( ξ j , ξ j (cid:48) ) , ( ξ k − j , ξ j (cid:48) − j ) , ( ξ k − j (cid:48) , ξ j − j (cid:48) ) .We turn now to the case when | α j,j (cid:48) | = 1 , i.e., c + d − . This occurs if, andonly if, | k and j (cid:48) = j + k or j (cid:48) = k . Suppose that | k and j (cid:48) = j + k . Then α j,j + k = − ξ j , and α l,l + k = α j,j + k if, and only if, l = j or l = j + k . In both cases, weget only the pair ( j, j + k ) . Now α k ,l = − ξ − l , so α k ,l = α j,j + k if, and only if, l = k − j or l = k − j . This shows that α j,j + k = α k − j, k = α k − j, k and the equality α j,j + k = α l,q holds only for these three pairs.(4) We can write α j,j (cid:48) in the form α j,j (cid:48) = 6(1 + (cid:60) ( ξ j ) + (cid:60) ( ξ j (cid:48) ) + (cid:60) ( ξ j (cid:48) − j )) + ξ j − j (cid:48) + ξ j + j (cid:48) + ξ j (cid:48) − j | ξ j + ξ j (cid:48) | . Therefore, α j,j (cid:48) is real if, and only if, ζ = ξ j − j (cid:48) + ξ j + j (cid:48) + ξ j (cid:48) − j is real. Setting31 = j − j (cid:48) and φ = j (cid:48) − j , we get (cid:61) ( ζ ) = sin( πθk ) + sin( πφk ) − sin( π ( θ + φ ) k ) . Then (cid:61) ( ζ ) = 0 ⇐⇒ sin( πθk ) + sin( πφk ) = sin( π ( θ + φ ) k ) ⇐⇒ π ( θ + φ ) k ) cos( π ( θ − φ ) k ) = 2 sin( π ( θ + φ ) k ) cos( π ( θ + φ ) k ) . Now, sin( π ( θ + φ ) k ) = 0 when j (cid:48) = k − j and cos( π ( θ − φ ) k ) = cos( π ( θ + φ ) k ) when j (cid:48) = 2 j or j = 2 j (cid:48) . Clearly, α j,k − j = 1 / (1 + ξ j + ξ k − j ) = α j, j . (5) As a a (cid:54) = 0 , we can write Ω kj,j (cid:48) = a a (1 − α j,j (cid:48) α ) , so Ω kj,j (cid:48) = 0 if, and onlyif, α j,j (cid:48) = 1 /α and the statement follows by (4).(6) When α is real, by (4), we have Ω j ,k − j = 0 when ξ j + ξ k − j + 1 = α . It followsthat ξ j = η with η = α − ± i (cid:113) − (cid:0) α − (cid:1) . Suppose that there exist a j for which ∆ kj = 0 , equivalently, ξ j + α − α − ξ j + 1 = 0 . Then ξ j + ξ k − j = − α − α − which gives ξ j = η with η = − α − α − ± i (cid:113) − (cid:0) α − α − (cid:1) . We have η = η if, and only if, α = 0 or .Then k = 3 p and j = p or p , which is excluded from our hypotheses, so η (cid:54) = η .The complex number η (resp. η ) is a k th -root of unity if, and only if, α isa root of the polynomial P ( α ) (resp. P ( α ) ) of degree k obtained by taking thenumerator of (cid:60) ( η k ) − (resp. (cid:60) ( η k ) − ). But if P and P have one common root,all the other roots must also be common. (This follows from the fact that the map cos θ = − ( α − / ( α − is a bijection for α ∈ ( −∞ , and cos θ = ( α − / is also abijection for α ∈ ( − , .) As P and P are distinct polynomials, it follows that theyhave no common roots. Consequently, ∆ kj (cid:54) = 0 for all j .The argument for showing that Ω kj,k − j (cid:54) = 0 when ∆ kj = 0 is the same as above.(7) We have, with the hypothesis, D j · D j (cid:48) = dim C O / (cid:104) xy, x + y (cid:105) = 4 . (cid:50) In view of Proposition 4.21 (7), we give in the rest of this section D j · D j (cid:48) for theexceptional branches only, i.e, when one or both branches have a singularity more de-generate than A or one or both of their components are tangential (which is equivalentto D j · D j (cid:48) > ). We start with the case when all the branches have an A -singularity. Proposition 4.22 Suppose that a = a = a = 0 , a a (cid:54) = 0 and ∆ kj (cid:54) = 0 for all j .Any k -folding map-germ F k satisfying the additional conditions in (a) or (b) is finitely A -determined and is topologically equivalent to one of the following germs: U kl : ( x, y ) (cid:55)→ ( x, y k , x y + 2 xy + y + y l − ) , l = 3 , . Every branch of the double point curve has an A -singularity and the invariants asso-ciated to the germs in these strata are as in Table 7.(a) If (cid:45) k and Ω kj,j (cid:48) (cid:54) = 0 for all distinct pairs, then F k is topologically equivalent to U k . (b) Suppose that k = 3 p and Ω kj,j (cid:48) (cid:54) = 0 for all distinct pairs with j, j (cid:48) (cid:54) = { p, p } . If a (cid:54) = 0 , then D p · D p = 5 and F k is topologically equivalent to U k . (b2) If a = 0 and CndU m (cid:54) = 0 , then D p · D p = 8 and F k is topologicallyequivalent to U k . Proof Each branch of the double point curve consists of a transverse intersection oftwo regular curves. In (a) all of these curves are pairwise transverse.In (b), j λ s ( x, y ) = x ( a x + a ϑ s y ) for s = p, p . Observe that a (cid:54) = 0 aswe supposed ∆ ks (cid:54) = 0 , so the branches D p and D p have one common line x = 0 intheir tangent cone and their associated curves tangent to this line are parametrisedby t (cid:55)→ ( γ s ( t ) , t ) with γ s ( t ) = − ( a / ( ϑ s a )) t + O (3) , so they have order of contact2 when a (cid:54) = 0 . Then D p · D p = dim C O / (cid:104) xy, x + y (cid:105) = 5 .When a = 0 , the two curves are parametrised by t (cid:55)→ ( γ s ( t ) , t ) with γ s ( t ) = − a a t + 2 ϑ s a ( a ( a a − a a ) + a a ) t + O (6) , and have contact order 5 when ( a ( a a − a a ) + a a (cid:54) = 0 , i.e., CndU m (cid:54) = 0 .Then D p · D p = dim C O / (cid:104) xy, x + y (cid:105) = 8 . For the topological normal forms, we choose a , a , a real with α > . ByProposition 4.21 (2), all ∆ kj are non-zero. Also, by Proposition 4.21 (4) and (5) Ω j,k − j = 0 when ξ j + ξ k − j + 1 = α , which cannot happen when α > , so the Ω kj,j (cid:48) in (a) and (b) are non-zero. As the strata are connected sets, we can choose a = a = 1 , a = 2 so that the conditions on ∆ kj and Ω kj,j (cid:48) are satisfied.The number of triple points is calculated as in Proposition 4.16. If k = 3 p , thegerm U k (resp. U k ) has T p, p = 3 (resp. T p, p = 6 ), while all other indices j and j (cid:48) give T j,j (cid:48) = 2 . (cid:50) Proposition 4.23 Suppose that a = a = a = 0 , a a (cid:54) = 0 , a (cid:54) = 0 and Ω kj ,j (cid:48) = 0 for some pair ( j , j (cid:48) ) , with j , j (cid:48) / ∈ { p, p } when k = 3 p . Then D s · D q = 5 for ( s, q ) ∈ { ( j , j (cid:48) ) , ( k − j , j (cid:48) − j ) , ( k − j (cid:48) , j − j (cid:48) ) } if, and only if, CndV m j ,j (cid:48) (cid:54) = 0 .For k = 3 p , we have D p · D p = 5 if, and only if, a (cid:54) = 0 . Any k -folding map-germ F k satisfying the above conditions is finitely A -determined and is topologically equivalentto V k,j ,j (cid:48) : ( x, y ) (cid:55)→ ( x, y k , x y + xy + α j ,j (cid:48) y + (cid:0) − β j ,j (cid:48) α j ,j (cid:48) (cid:1) x y + y ) . The invariants associated to germs in these strata are as in Table 7. Proof When a = 0 and k = 3 p , we have ∆ kp = ∆ k p = 0 . This case is dealt within Proposition 4.25. With the hypothesis and Proposition 4.21(5), we have Ω j ,j (cid:48) =Ω k − j ,j (cid:48) − j = Ω k − j (cid:48) ,j − j (cid:48) = 0 . We need to consider the order of contact between the two tangential componentsof the pairs ( D s , D q ) with ( s, q ) ∈ { ( j , j (cid:48) ) , ( k − j , j (cid:48) − j ) , ( k − j (cid:48) , j − j (cid:48) ) } . Using33he fact that if two quadratic equations x + a i x + b i = 0 , i = 1 , , have one root incommon, then the root is given by x = − ( b − b ) / ( a − a ) , we can get the initialterms of parametrisations of the tangential components of the above pairs. These aregiven by t (cid:55)→ ( t, γ l ( t )) , l = s, q , with γ l ( t ) = − a a (cid:0) ξ s + ξ q (cid:1) t + λ l t + O (3) , l = s, q. A calculation shows that λ s − λ q = 0 if, and only if, CndV m s,q = a a β s,q + a a (2 a a − a a ) α s,q + a a = 0 , with β j,j (cid:48) = ξ j +3 j (cid:48) + ξ j + j (cid:48) + ξ j (cid:48) + 2 ξ j + j (cid:48) + 2 ξ j +2 j (cid:48) + ξ j + 2 ξ j + j (cid:48) + ξ j + ξ j (cid:48) (1 + ξ j + ξ j (cid:48) ) . (8)Observe that β j,j (cid:48) = β k − j,j (cid:48) − j = β k − j (cid:48) ,j − j (cid:48) , so CndV m s,q has the same valuefor ( s, q ) ∈ { ( j , j (cid:48) ) , ( k − j , j (cid:48) − j ) , ( k − j (cid:48) , j − j (cid:48) ) } . It follows that D s · D q =dim C O / (cid:104) xy, x + y (cid:105) = 5 if, and only if, CndV m j ,j (cid:48) (cid:54) = 0 .For the topological model, we take a = a = 1 and a = α j ,j (cid:48) so Ω j ,j (cid:48) = 0 .We also set a = 1 , a = a = 0 , then CndV m j ,j (cid:48) = β j ,j (cid:48) + a α j ,j (cid:48) . We set a = 1 − β j ,j (cid:48) /α j ,j (cid:48) , so that CndV m j ,j (cid:48) (cid:54) = 0 . For calculating the triple points, wefind that T j,j (cid:48) = 3 for all j and j (cid:48) with j (cid:54) = j (cid:48) . (cid:50) We deal now with the case when one pair of branches of the double point curvehas a singularity more degenerate than A (see Proposition 4.21(1)). Remark 4.24 For k = 3 p , the singularities U k and V k,j,j (cid:48) have the same invariants,but the intersection numbers between their double point branches shows that they arenot topologically equivalent. Proposition 4.25 Suppose that a = a = a = 0 , a a (cid:54) = 0 . The strata beloware of codimension , and the invariants associated to germs in the strata are as in Table 7.(1) Suppose that ∆ kj = ∆ kk − j = 0 , with j / ∈ { p, p } when k = 3 p ( so a (cid:54) = 0) .The branches D j and D k − j have an A -singularity if, and only if, CndW A (cid:54) = 0 . Wehave D j · D j = D k − j · D j = 4 for all distinct pairs. When k = 3 p and a (cid:54) = 0 , we have D p · D p = 5 . Any k -folding map-germ F k satisfying the above conditions is A -finitelydetermined and is topologically equivalent to W k,j : ( x, y ) (cid:55)→ ( x, y k , x y + xy + ( ξ j + 1) ξ j + ξ j + 1) y + 4 x y + y ) . (2) If k = 3 p and j = p , then D p and D p have an A -singularity if, and only if, a (cid:54) = 0 . We have D p · D p = 8 if, and only if, Cnd (cid:102) W m (cid:54) = 0 , and D s · D j = 5 for s = p, p and j (cid:54) = s . Any k -folding map-germ F k satisfying the above conditions is A -finitely determined and is topologically equivalent to W p,p : ( x, y ) (cid:55)→ ( x, y k , x y + y + y + y ) . roof The condition for an A -singularity of D j and D k − j follows by analysing the3-jets of, respectively, λ j and λ k − j . Observe that when | k , ξ j = − is a solutionof ∆ kj = ∆ kk − j = 0 if, and only if, a a = 0 . As we are assuming a a (cid:54) = 0 , we have j (cid:54) = k/ in (1) when | k .With the hypothesis in (1), D j · D j = D k − j · D j = dim C O / (cid:104) x + y , x − y (cid:105) = 4 for any distinct pairs.For (2), we have D p · D p = dim C O / (cid:104) x + y , y (cid:105) = 8 when Cnd (cid:102) W m (cid:54) = 0 .For the topological normal forms, we use the fact that ∆ kj = 0 if and only if α = a / ( a a ) = 4( ξ j + ξ j + 1) / ( ξ j + 1) (see the proof of Proposition 4.21) andset a = a = 1 , so a = ( ξ j + 1) / ξ j + ξ j + 1) . We have T j,j (cid:48) = 2 for all j and j (cid:48) with j (cid:54) = j (cid:48) . (cid:50) Proposition 4.26 Suppose that a = a = a = 0 , a = 0 , a a (cid:54) = 0 . All thebranches of the double point curve share only one line in their tangent cones exceptwhen k = 3 p where the branches D p and D p have the same tangent cone. Any k -folding map-germ F k satisfying the additional conditions in (a) , (b) , (c) or (d) isfinitely A -determined and is topologically equivalent to X k : ( x, y ) (cid:55)→ ( x, y k , xy + y + x y + y ) . The stratum is of codimension , and the invariants associated to the germs in thestrata above are as in Table 7.(a) If (cid:45) k , (cid:45) k , all of the branches D j of the double point curve have an A -singularity and D j · D j (cid:48) = 5 , j (cid:54) = j (cid:48) , when a (cid:54) = 0 . (b) If k = 2 p , (cid:45) k , the branches D j , j (cid:54) = p , behave as in (a) . The branch D p has an A -singularity if, and only if, a (cid:54) = 0 . We also have D j · D p = 5 , j (cid:54) = p . (c) If k = 3 p , (cid:45) k , the branches D j behave as in (a) except for D p and D p where D p · D p = 6 when a a (cid:54) = 0 . (d) If k = 6 p , when a a (cid:54) = 0 , the exceptional branches of the double point curvebehave as in (b) and (c) , the remaining branches behave as in (a) . Proof We have j λ j ( x, y ) = y ( ϑ j a x + ϑ j a y ) . For (a), ϑ j (cid:54) = 0 for all j and ϑ j ϑ j (cid:54) = ϑ j (cid:48) ϑ j (cid:48) for j (cid:54) = j (cid:48) , so all of the branches of the double point curve have an A -singularityand one common line y = 0 in their tangent cone. The component of D j with tangent y = 0 can be parametrised by t (cid:55)→ ( t, γ j ( t )) with by γ j ( t ) = − a / ((1 + ξ j ) a ) t + O (3) .Therefore, when a (cid:54) = 0 , we have D j · D j (cid:48) = dim C O / (cid:104) xy, y + x (cid:105) = 5 .The remaining parts of the proof follow similarly and are omitted. (cid:50) roposition 4.27 Suppose that a = a = a = 0 , a = 0 , a a (cid:54) = 0 . All thebranches of the double point curve share only one line in their tangent cones. Any k -folding map-germ F k satisfying the additional conditions in (a) or (b) is finitely A -determined and is topologically equivalent to Y k : ( x, y ) (cid:55)→ ( x, y k , − xy + x y + y + y ) . The stratum is of codimension , and the invariants associated to germs in thestratum are as in Table 7.(a) If (cid:45) k , all of the branches of the double point curve have an A -singularity and D j · D j (cid:48) = 5 , j (cid:54) = j (cid:48) , when a (cid:54) = 0 . (b) If k = 2 p , the branch D p has an A -singularity if, and only if, CndY A (cid:54) = 0 .Then D p · D j = 6 , j (cid:54) = p , and D j · D p + j = 6 , ≤ j < p , if and only if CndY m j (cid:54) = 0 . The other pairs of branches behave as in (a) . Proof We have j λ j ( x, y ) = x ( a x + a ϑ j y ) so when (cid:45) k , all of the branches ofthe double point curve have an A -singularity as ϑ j (cid:54) = 0 for all j . All the brancheshave only x = 0 as a common line in their tangent cone. The components of thebranches which are tangent to x = 0 are parametrised by t (cid:55)→ ( γ j ( t ) , t ) , with γ j ( t ) = − a (1 + ξ j ) /a t + O (3) . When a (cid:54) = 0 , D j · D j (cid:48) = dim C O / (cid:104) xy, x + y (cid:105) = 5 .When k = 2 p , we have ϑ p = ϑ p = 0 , so considering the 4-jet of λ p , we find thatthe branch D p has an A -singularity if, and only if, CndY A (cid:54) = 0 . When this is thecase, D p · D j = dim C O / (cid:104) xy, x + y (cid:105) = 6 , for j (cid:54) = p .All pairs of branches D j , D p + j , ≤ j < p have tangential component parametrisedby t (cid:55)→ ( γ s ( t ) , t ) , with γ s ( t ) = − a (1 + ξ j ) /a t + α s t + O (4) , s = j, p + j , with α − α (cid:54) = 0 if and only if CndY m j (cid:54) = 0 . When this is the case, D j · D p + j =dim C O / (cid:104) xy, x + y (cid:105) = 6 , for ≤ j < p . The other pairs of branches behave aswhen (cid:45) k . (cid:50) Remark 4.28 The stratification S k is determined, for each k ≥ , by the conditionsin Tables 2, 4 and 6 ( S follows from the results in [10] and S is indicated in Table1). Each stratum corresponds to a single topological class and all the topologicalclasses listed in Tables 2, 4 and 6 are topologically distinct. Indeed, as pointed outin Remark 4.17 (resp. Remark 4.19 and resp. Remark 4.24) Q k and (cid:101) Q k (resp. Q and R and resp. U k and V k,j,j (cid:48) ) are not topologically equivalent. For the remaingclasses, it follows by comparing the invariants in Tables 3, 5 and 7, that they are alltopologically distinct. k -folding map-germs We define, by varying the plane π ∈ Graff(2 , , the family of Whitney k -folds Ω k : C × Graff(2 , → C , given by Ω k ( p, π ) = ω kπ ( p ) , with ω kπ as in Definition 2.2.36iven a complex surface M in C , we call the restriction of Ω k to M the family of k -folding maps on M and denote it by F k . We have F k ( p, π ) = F πk ( p ) = ω kπ ( p ) for all p ∈ M and π ∈ Graff(2 , .Recall that a property of surfaces is said to be generic if it is satisfied in a residualset of embeddings of the surfaces to C . The image of a surface M by an embeddingin the residual set is then called generic, or simply that M is generic. When k is large,the A -singularities of F πk may have high A e -codimensions (for the cases in this paper,this means high modality). However, they do occur on generic surfaces. To make senseof this, we follow a similar approach to that in [6] and proceed as follows.As we are interested here in the local singularities of k -folding maps, we considerthe setting in Remarks 2.3(4) at a point p ∈ M and choose a suitable system ofcoordinates so that π : y = 0 and F k = F π k ( x, y ) = ( x, y k , f ( x, y )) for ( x, y ) in asmall enough neighbourhood U of the origin. A plane π = ( d, v ) near π is obtainedby applying a translation T π followed by an orthogonal transformation R π ∈ U (3) to π . We choose T π and R π as follows. The translation T π takes the origin to the pointof intersection of π with the y -axis (the point exists because π is orthogonal to the y -axis and π is close to π ). The transformation R π near the identity (and is takento be the identity if v is parallel to v ) takes v/ || v || to v = (0 , , and fixes the linethrough the origin orthogonal to v and v . By varying the planes π in a neighbourhood V of π in Graff(2 , , we get the family of k -folding maps given by F k : U × V → C ,with F k (( x, y ) , π ) = ( R π ◦ T π ) ◦ ω k ◦ ( R π ◦ T π ) − ( φ ( x, y )) , with φ ( x, y ) = ( x, y, f ( x, y )) . (We choose R π and T π to depend analytically on π .)The A -type of the singularity of F πk at a given point in U is the same as that of thegerm of ˜ F k (( x, y ) , π ) = ˜ F kπ ( x, y ) = ( R π ◦ T π ) − ◦ F k (( x, y ) , π ) = ω k ◦ ( R π ◦ T π ) − ( φ ( x, y )) at that point. At any point p = ( x, y ) ∈ U , there exist a bi-holomorphic map-germ K : ( C , → ( C , ( x, y )) such that ( R π ◦ T π ) − ( φ ( K ( X, Y )) = (¯ x + X, ¯ y + Y, ¯ z + g p ( X, Y )) , for some germ of a holomorphic function g p , where (¯ x, ¯ y, ¯ z ) = ( R π ◦ T π ) − ( φ ( x, y )) . Composing ˜ F kπ with K gives the germ ( ˜ F kπ ) p of ˜ F kπ at p ( ˜ F kπ ) p ( X, Y ) = (¯ x + X, (¯ y + Y ) k , ¯ z + g p ( X, Y )) ∼ A ( X, (¯ y + Y ) k , g p ( X, Y )) . Observe that g p depends on the choice of R π , T π and of the coordinates system,but the A -class of the resulting germs ( ˜ F kπ ) p is independent of these choices.37learly, a necessary condition for ( ˜ F πk ) p to be singular at the origin is ¯ y = 0 ,equivalently, p ∈ π . We define the family of maps Φ : U × V → C × J l (2 , given by Φ( p, π ) = ( (cid:104) v, p (cid:105) − d, j l g p (0 , . The map Φ plays a similar role to the Monge-Taylor map in [6]. Here we include thefirst component to capture the planes through a given point p ∈ M which can giverise to singular k -folding map-germs at p .We stratify C × J l (2 , by × S k , together with C × J l (2 , \ × J l (2 , . Fol-lowing standard transversality arguments (see for example [6]), one can show that fora residual set of local embeddings of M in C , the map Φ is transverse to the stratain C × J l (2 , . As the domain of the family Φ is of dimension 5, for a generic localembedding of M in C , Φ intersects a stratum × X only when X has codimension ≤ in J l (2 , . This means that the only singularities of k -folding maps that canoccur on a generic surface are those belonging to the strata listed in Tables 1, 2, 4,6. Furthermore, Φ is transverse to these strata. Therefore, the locus of points wherethe l -jets of the k -folding map-germs belong to a stratum X of codimension (resp. ) is a regular curve on M (resp. isolated points on this curve). The above discussionalso clarifies why the singularities of F πk occur generically even though they may havehigh A e -codimensions: they belong to strata of low codimension. (The strata can beviewed as a gluing of non A -equivalent orbits which depends on a finite set of moduli.)It is worth observing that the above discussion is valid for smooth surfaces in R . k -folding map-germs on surfaces in R We consider in this section the geometry of k -folding maps-germs on smooth (i.e.,regular and of class C ∞ ) surfaces in R . Robust features on a surface in R is a terminology introduced by Ian Porteous toindicate special characteristic features that can be traced when the surface evolves.What is sought after in applications are robust features which are represented by curvesor points on the surface as these form a “skeletal structure” of the surface (open regionsbounded by robust curves are also robust features). They play an important role incomputer vision and shape recognition (see for example [27]) as they can be used todistinguish two shapes (surfaces) from each other and, in some cases, reconstruct thesurface.We consider here the parabolic, ridge, sub-parabolic and flecnodal curves on asmooth surface M in R and special points on these curves (see [15] for referenceson work on these curves from a singularity theory point of view). These are robustfeatures on M . We recall briefly what they are.38 he parabolic curve is the locus of points where the Gaussian curvature vanishes.It is captured by the contact of the surface M with planes: it is the locus of pointswhere the height function along a normal direction to M has an A ≥ -singularity. Theparabolic curve is regular on a generic surface, and the height function has an A -singularity at its points except at isolated ones, called Cusps of Gauss , where it hasan A -singularity. The ridge is the locus of points on M where a principal curvature is extremal alongits associated lines of principal curvature. It is also the locus of points on M whichcorrespond to singular points on its focal set. The ridge is captured by the contact of M with spheres: it is the locus of points on M where the distance squared functionfrom a given point in R (the point belong to the focal set) has an A ≥ -singularity.Away from umbilic points (i.e., points where the principal curvatures coincide), theridge is a regular curve on a generic surface and the distance squared function has an A -singularity at its points except at isolated points where it has an A -singularity.At umbilics, the ridge consist of one regular curve or a transverse intersection of threeregular curves. (Umbilics and A -points are used as seed points for drawing ridges ona given shape, see [27].)The sub-parabolic curve is the locus of points on M corresponding to parabolicpoints of its focal set. It is the locus of geodesic inflections of the lines of principalcurvature; it is also the locus of points along which a principal curvature is extremalalong the other lines of curvature. It is captured by the singularities of the 2-foldingmap on M : it is the locus of points where some map F π has an S ≥ -singularity (or onewhich is adjacent to an S -singularity). The singularities of F and their geometriccharacterisation on a generic surface parametrised in Monge form z = f ( x, y ) at a givennon-umbilic point, with f as in (2), are as in Table 8. At umbilics, the sub-paraboliccurve consist of one regular curve or a transverse intersection of three regular curves.Observe that the ridge is also captured by the singularities of F π (see Table 8).The flecnodal curve is the locus of geodesic inflections of the asymptotic curves. Itis captured by the contact of the surface M with lines: it is the locus of points wherethe orthogonal projection of the surface has a singularity of type swallowtail or worse.We recall briefly some results on these projections as they are needed for interpretingthe singularities of k -folding maps.The orthogonal projection P v of M along the direction v ∈ S to the plane T v S is given by P v ( p ) = p − ( p · v ) v , with p ∈ M . This can be represented locally bya map-germ from the plane to the plane. Varying v yields the family of orthogonalprojections P : M × S → T S given by P ( p, v ) = ( v, P v ( p )) . A transversality theorem asserts that for an open and dense (i.e., generic) set ofembeddings φ : U → R , the surface M = φ ( U ) has the following property: for any v ∈ S the map-germ P v has only local singularities A -equivalent to one in Table 9 atany point on M . By translating p ∈ M to the origin and taking M locally at p inMonge form, an open subset of M is parametrised by φ ( x, y ) = ( x, y, f ( x, y )) , with f F . Name Algebraic conditions and geometric meaningCrosscap a (cid:54) = 0 B = S a = 0 , a (cid:54) = 0 , a (cid:54) = 0 General smooth point of focal set B a = 0 , a (cid:54) = 0 , a = 0 , CndN A = 4 a a − a (cid:54) = 0 General cusp point of focal set corresponding to a point on the ridgecurve B a = 0 , a (cid:54) = 0 , a = 0 , CndN A = 0 , CndN A = 8 a a − a a a + 2 a a a − a a (cid:54) = 0 (Cusp) point of focal set in closure of parabolic curve on symmetry set S a = 0 , a = 0 , a (cid:54) = 0 , a (cid:54) = 0 Parabolic smooth point of focal set corresponding to a point on thesub-parabolic curve S a = 0 , a = 0 , a (cid:54) = 0 , a = 0 , a (cid:54) = 0 Cusp of Gauss at smooth point of focal set C a = 0 , a = 0 , a = 0 , a (cid:54) = 0 , a (cid:54) = 0 Intersection point of cuspidal-edge and parabolic curve on focal set having no constant nor linear terms. Projecting along the direction v = (0 , , gives P v ( x, y ) = ( x, f ( x, y )) , which is singular at the origin. Table 9 shows the conditionson the coefficients of f for the map-germ P v to have a singularity at the origin of A e -codimension ≤ , where d e ( G, A ) denotes the A e -codimension of G (see for example[15]),The projection P v has a fold singularity at p if, and and only if, v is a non-asymptotic tangent direction to M at p . The singularity at p is of type cusp or worseif, and only if, v is an asymptotic direction at p . For a generic surface M , the closureof the set of points where P v has a swallowtail (resp. lips/beaks) singularity is aprecisely the flecnodal (resp. parabolic) curve. The flecnodal curve meets tangentiallythe parabolic curve at the cusps of the Gauss map, which are the gulls singularities of P v [2]. We call a point where this happens a gulls-point of M The goose (resp. butterfly) singularities of P v appear at special points on theparabolic (resp. flecnodal) curve. We call a point on M where these singularitiesoccur a goose-point (resp. butterfly-point ) on M .The above robust features are defined in terms of the principal curvatures κ and κ . We can suppose κ < κ (away from umbilic curves) and give different colors tothe robust features associated to each principal curvature. For instance, we can havea blue ridge (associated to κ ) and a red ridge (associated to κ ).40able 9: A e -codimension ≤ singularities of map-germs G : ( R , → ( R , . Name Normal form d e ( G, A ) Algebraic conditions on f in (2)for the singularities of G ( x, y ) = ( x, f ( x, y )) Fold ( x, y ) a (cid:54) = 0 Cusp ( x, xy + y ) a = 0 , a (cid:54) = 0 , a (cid:54) = 0 Swallowtail ( x, xy + y ) a = 0 , a = 0 , a (cid:54) = 0 , a (cid:54) = 0 Lips/beaks ( x, y ± x y ) a = 0 , a = 0 , a (cid:54) = 0 , a − a a (cid:54) = 0 Goose ( x, y + x y ) a = 0 , a = 0 , a − a a = 0 , a (cid:54) = 0 , a a − a a a + 9 a a a − a a (cid:54) = 0 Butterfly ( x, xy + y ± y ) a = 0 , a = 0 , a = 0 , a (cid:54) = 0 , a (cid:54) = 0 , (8 a a − a ) a + 2 a ( a a − a a ) a +35 a a (cid:54) = 0 Gulls ( x, xy + y + y ) a = 0 , a = 0 , a = 0 , a (cid:54) = 0 , a (cid:54) = 0 , a a − a a a + 4 a a (cid:54) = 0 k -folding map-germs We obtain here the robust features determined by the singularities of k -folding map-germs, for k ≥ . We start with the case where p is not an umbilic point and treat the k = 3 case separately. Theorem 6.1 Let M be a generic surface in R , p a non umbilical point on M and π a plane through p and orthogonal to v . The map-germ F π at p is an immersion if, andonly if, v is not tangent to M at p . It has an S -singularity if, and only if, v ∈ T p M but is neither a principal nor an asymptotic direction. We have the following when v is a principal or an asymptotic direction at p . (1) If v is a principal direction at p , then the possible singularities of F π at p andtheir geometric characterisations are as follows: S : p is not on the sub-parabolic curve associated to v . S : p is a generic point on the sub-parabolic curve associated to v . S : p is an S -point on the sub-parabolic curve associated to v . (2) Suppose that v is an asymptotic and that p is a hyperbolic point. Then F π has asingularity at p of type H k , k = 2 , , . The H -singularities occur on a regularcurve on M . We call its closure the H -curve . The H -singularities occur atisolated points on this curve. (3) If p is a parabolic point and v is the unique asymptotic direction at p which isalso a principal direction, then the possible topological classes of F π at p andtheir geometric characterisations are as follows: : p is a generic point on the parabolic curve. U : p is an A ∗ -point ( see [11, 12]) ; it is on the closure of the H -curve. X : p is on the sub-parabolic curve associated to v ; the principal map withvalue v at p has a beaks singularity at p ([9]) . W , : p is on the sub-parabolic curve associated to the other principal di-rection v ⊥ ; it is on the closure of the H -curve; the frame map has across-cap singularity at p ([9]) . Proof The proof follows by considering the defining equations and the open condi-tions of the strata of -folding map-germs in Theorem 4.4 and the geometric interpre-tation of the algebraic conditions in Table 1 given in Tables 8 and 9. (cid:50) Proposition 6.2 The H -curve of a generic surface is a regular curve and meets theparabolic curve tangentially at A ∗ and W , -points. Proof The regularity of the H -curve follows by a transversality argument. Wecompute the 1-jets of the parabolic and H -curves and find that they are tangentialat their points of intersection. This is expected as the H -singularities of F π occurwhen v is an asymptotic direction, so the H -curve lies in the closure of the hyperbolicregion of the surface. (cid:50) We turn now to the case k ≥ , still assuming p not to be an umbilical point. Werecall that some topological classes in Tables 2, 4, 6 come with divisibility conditionson k . For example, the N k class requires | k (see Remarks 6.4). Theorem 6.3 Let k ≥ and let M be a generic surface in R , p a non umbilicalpoint on M and π a plane through p and orthogonal to v . The map-germ F πk at p isan immersion if, and only if, v is not tangent to M at p . It has a singularity whichis topologically equivalent to M k if, and only if, v ∈ T p M but is neither a principalnor an asymptotic direction at p . We have the following when v is a principal or anasymptotic direction at p . (1) If v is a principal direction at p , then F πk belongs to a stratum in Table 2 . Thepossible topological classes of F πk and their geometric interpretations are as fol-lows: M k : p is not on the sub-parabolic curve associated to v . M k : p is a generic point on the sub-parabolic curve associated to v . M k : p is an S -point on the sub-parabolic curve associated to v . N k : p is a generic point on the ridge curve associated to v . N k : p is a B -point on the ridge curve associated to v . O k : p is a C -point ( intersection point of the ridge and sub-parobolic curvesassociated to v ) . If p is a hyperbolic point and v is an asymptotic direction at p , then F πk belongs toa stratum in Table 4 . The possible topological classes of F πk and their geometricinterpretations are as follows: P k : p is not on the flecnodal curve associated to v , and is not on the H -curve associated to v when | k . P k : p is a generic point on the H -curve associated to v . P k : p is an H -point on the H -curve associated to v . Q k : p is a generic point on the flecnodal curve associated to v . Q k : p is a point of intersection of the flecnodal and H -curves associated to v . (cid:101) Q k : p is a special point on the flecnodal curve associated to v . R k : p is a butterfly-point on the flecnodal curve associated to v . (3) If p is a parabolic point and v is an asymptotic ( and a principal ) direction at p ,then F πk belongs to a stratum in Table 6 . The possible topological classes of F πk and their geometric interpretations are as follows: U k : p is a generic point on the parabolic curve. U k : p is an A ∗ -point. V k,j,j (cid:48) : p is a special point on the parabolic curve. W k,j : p is a special point on the parabolic curve. W q,q : p is on the sub-parabolic curve associated to v ⊥ ; it is on the closure ofthe H -curve associated to v ; the frame map has a cross-cap singularityat p ([9]) . X k : p is on the intersection of the parabolic and sub-parabolic curves asso-ciated to v ; the principal map with value v at p has a beaks singularityat p ([9]) . Y k : p is a cusp of Gauss point and a gulls-point. Proof With the setting as in the proof of Theorem 6.1, the results follow by inter-preting the conditions in Tables 2, 4 and 6 using Tables 8 and 9. (cid:50) Remarks 6.4 1. The singularities in Branches 1 and 2 are associated to principaldirections, those in Branch 3 to asymptotic directions at hyperbolic points and thosein Branch 4 to asymptotic directions at parabolic points.2. Theorems 6.1 and 6.3 show clearly that k -folding maps capture the robustfeatures obtained by -folding maps and by the contact of the surface with lines,planes and spheres, giving thus new geometric characterisations of these features anda unified approach to study them. We also obtain a new 1-dimensional robust feature(the H -curve) and several -dimensional ones: the H , Q k , (cid:101) Q k , V k,j,j (cid:48) and W k,j points.3. The H -curve is captured by k -folding maps when k is divisible by . Thesub-parabolic and flecnodal curves are captured by k -folding maps for any k while the43idge curve is captured by k -folding maps when k is even. Regarding the ridge, it isthe locus of points where the surface has more infinitesimal symmetry with respect toplanes ([10]). The map-germs ( x, y ) (cid:55)→ ( x, y p , f ( x, y )) identify the pair of points ( x, y ) and ( x, − y ) , which explains why all the F p folding maps capture the ridge curve.4. The condition ∆ kj = 0 can be satisfied for any k, j when the coefficients a sl arereal, so W k,j -points can occur on surfaces in R for all k, j . For the V k,j,j (cid:48) topologicalclass, it follows from Proposition 4.21 that only V k,j,k − j , V k,j, j and V k,k − j, j -pointscan occur on surfaces in R .5. Theorem 6.1 gives a new geometric interpretation for the A ∗ -points in [11, 12].6. Observe that the open conditions (those involving expressions (cid:54) = 0 ) for thesingularities of the k -folding map in Tables 1, 2, 4, 6 and their associated ones inTables 8 and 9 are not always identical. For the -dimensional robust features, theopen conditions in both tables are satisfied on a generic surface. For the -dimensionalrobust features, this means that some special points in one setting are not special inthe other. For example, the -folding map does not distinguish between a C -pointand a generic point on the sub-parabolic curve.We consider now the situation at umbilic points. For a generic surface M , theseoccur at isolated points in its elliptic region, and every direction in the tangent planeof M at such points can be considered a principal direction. We take M locallyin Monge form z = f ( z, y ) , consider the origin to be an umbilic point and write f ( x, y ) = κ ( x + y ) + C ( x, y ) + O ( x, y ) where C is a homogeneous cubic form in x, y .We can take C ( x, y ) to be the real part of z + βz z , with z = x + iy and β = s + it (see for example [10]). Then, C = (1 + s ) x − tx y + ( s − xy x − ty . Theorem 6.5 Let k ≥ and let M be a generic smooth surface in R , p an umbilicpoint on M and π a plane through p and orthogonal to v ∈ T p M .1. If (cid:45) k , then for almost all directions v in T p M the singularity of F πk at p is oftype S when k = 3 and of type M k when k ≥ . There are three directions ( resp.one direction ) where the singularity is of type S when k = 3 and of type M k when k ≥ if β is inside ( resp. outside ) the outer hypocycloid β = − e iθ + e − iθ ) in Figure 4 .2. If | k , then for almost all directions v in T p M the singularity of F πk at p is oftype M k . There are three directions ( resp. one direction ) where the singularityis of type M k if β is inside ( resp. outside ) the hypocycloid β = − e iθ + e − iθ ) .There are also three directions ( resp. one direction ) where the singularity is oftype N k when β is inside ( resp. outside ) the inner hypocycloid β = 2 e iθ + e − iθ in Figure 4 . Proof We take v = (cos( θ ) , sin( θ ) , , θ ∈ [0 , π ] and consider the rotation R = sin( θ ) cos( θ ) 0 − cos( θ ) sin( θ ) 00 0 1 , which takes the direction (0 , , to v . Then, F πk ◦ R − ( x, y ) = ( x sin( θ ) − y cos( θ ) , ( x cos( θ ) + y sin( θ )) k , f ( x, y )) . Changes of coordinates in the source give F πk ◦ R − ( X, Y ) = ( X, Y k , f ( X sin( θ ) + Y cos( θ ) , − X cos( θ ) + Y sin( θ ))) . We denote by ¯ a lj the coefficient of X l − j Y j in the Taylor expansion of f ( X sin( θ ) + Y cos( θ ) , − X cos( θ ) + Y sin( θ )) . The proof follows then considering the conditions forthe singularities of F πk in Tables 1 and 2.(1) We have ¯ a = ( s − 3) cos( θ ) − t cos( θ ) sin( θ ) + (9 + s ) sin( θ ) cos( θ ) − t sin( θ ) . When (cid:45) k and ¯ a (cid:54) = 0 , the singularity of F πk at the origin is of type S when k = 3 or of type M k when k ≥ .The coefficient ¯ a is a cubic form in cos( θ ) and sin( θ ) . Its discriminant is thehypocycloid β = − e iθ + e − iθ ) . The cubic has three roots for β inside the hypocy-cloid and one root when it is outside. For v corresponding to one of these roots, thesingularity of F πk is of type S when k = 3 and of type M k when k ≥ , provided ¯ a (cid:54) = 0 . The condition ¯ a (cid:54) = 0 is satisfied at umbilic points on generic surfaces.(2) We have ¯ a = (1 + s ) cos( θ ) − t cos( θ ) sin( θ ) + ( s − 3) cos( θ ) sin( θ ) − t sin( θ ) . | k (so k ≥ ) and ¯ a (cid:54) = 0 , the singularity of F πk at the origin is of type M k . We also get the M k singularities as in (1) when ¯ a = 0 .The coefficient ¯ a is also a cubic form in cos( θ ) and sin( θ ) . Its discriminantis the hypocycloid β = 2 e iθ + e − iθ . The cubic has three roots for β inside thehypocycloid and one root when it is outside. For v corresponding to one of theseroots, the singularity of F πk is of type N k if ¯ a (cid:54) = 0 .We have ¯ a = ¯ a = 0 if, and only if, β is on one of the tangent lines t (3 s − t ) = 0 to the hypocycloids at their cusp points, see Figure 4. (On these lines, the singularityis of type O k or more degenerate. This singularity does not occur at umbilic pointson a generic surface.) (cid:50) Remark 6.6 1. Theorem 6.5 is merely another interpretation of the results in [10, 32]when using the geometric characterisations of the singularities of k -folding maps inTheorems 6.1 and 6.3. We know from [6, 10, 32] that there are one or three ridgecurves and one or three sub-parabolic curves at umbilic points on a generic surface.These curves meet transversally and change colour at the umbilic point.2. Figure 4 is first obtain in [10] when considering 2-folding maps. In that caseboth hypocycloids are present, whereas when k ≥ only one of them is present when k is odd (both are present when k is even). Also in [10] is considered the circle | β | = 3 which corresponds to the Monge-Taylor map failing to be transverse to the umbilicsstratum. The circle | β | = 1 is also exceptional and corresponds to two ridges throughthe umbilic being tangential, see [8]. As these conditions are geometric, the circles | β | = 1 and | β | = 3 can also be considered exceptional for k -folding maps and areadded to Figure 4.3. Umbilic points on a generic surface occur at elliptic points, that is why we donot get flecnodal curves or H -curves through such points. Acknowledgments. GPS was partially supported by the Basque Government throughthe BERC 2018-2021 program and Gobierno Vasco Grant IT1094-16, by the SpanishMinistry of Science, Innovation and Universities: BCAM Severo Ochoa accreditationSEV-2017-0718, by the ERCEA Consolidator Grant 615655 NMST, and by Programade Becas Posdoctorales en la UNAM, DGAPA, Instituto de Matemáticas, UNAM.FT was partially supported by the FAPESP Thematic project grant 2019/07316-0and the CNPq research grant 303772/2018-2. References [1] V. I. Arnold, S. M. Guse˘ın-Zade and A. N. Varchenko, Singularities of differen-tiable maps. Vol. I. 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