Transversal special parabolic points in the graph of a polynomial obtained under Viro's patchworking
Fuensanta Aroca Bisquert, Angelito Camacho Calderón, Mirna Gómez Morales
TTransversal special parabolic points in the graph of apolynomial obtained under Viro’s patchworking ∗ Fuensanta Aroca, Angelito Camacho-Calder´on, Mirna G´omez-Morales.Unidad Cuernavaca del Instituto de Matem´aticas,Universidad Nacional Aut´onoma de M´exico.UMI - “Laboratorio Solomon Lefschetz” CNRSOctober 18, 2018
Abstract
In this article we focus on the study of special parabolic points in surfaces arising asgraphs of polynomials, we give a theorem of Viro’s patchworking type to build familiesof real polynomials in two variables with a prescribed number of special parabolic pointsin their graphs. We use this result to build a family of degree d real polynomials in twovariables with ( d − d −
9) special parabolic points in its graph. This brings the numberof special parabolic points closer to the upper bound of ( d − d −
12) when d ≥ Points in a surface immersed in a 3-dimensional affine space are classified in terms of the contactorder of their tangent lines to the surface. On generic surfaces, parabolic points appear alonga curve which separates the hiperbolic domain from the elliptic domain and, among parabolicpoints, there are points where the highest contact order is reached in the direction of its onlyasymptotic line. These points are called special parabolic points or Gaussian cusps .Finding the number of special parabolic points in the graph of a generic polynomial of degree d , has been of special interest for the last century. For example, in [8], an upper bound of2 d ( d − d −
24) special parabolic points in generic algebraic surfaces of degree d in RP isgiven. In [12], A. Ortiz-Rodr´ıguez builds a family of polynomials whose graphs describe genericsurfaces with d ( d −
2) special parabolic points. And in [5], together with Hern´andez-Mart´ınezand S´anchez-Bringas, she proves that there are at most ( d − d −
12) special parabolic pointsin the graph of a polynomial of degree d . ∗ Research partially supported by PAPIIT-UNAM IN108216, ECOS M14M03, LAISLA, CONACYT (Mexico)grants 225387-292689 and 224855-291053. a r X i v : . [ m a t h . AG ] M a y iro’s patchworking was introduced in the late seventies [16] as a technique to glue simple alge-braic curves in order to construct real algebraic non-singular curves with prescribed topology.Details on this technique can be found for example in [17]. Among its many applications, it hasbeen used by E. Brugall´e and B. Bertrand to construct examples of real algebraic hypersurfacesin the projective plane with ( d − compact connected components in their parabolic curves[2], and by E. Brugall´e and L. L´opez de Medrano to construct examples of real algebraic curvesin the projective plane with the maximum number of real inflection points [3].In this article, we glue simple graphs in order to build a new graph with a prescribed numberof special parabolic points in it. Our main result is a theorem of Viro’s patchworking type: Theorem 8.6 (Viro’s Theorem for transversal special parabolic points) Let ∆ ⊂ R be apolyhedron with vertices in Z and let τ be the convex polyhedral subdivision of ∆ induced by λ : ∆ → R ≥ . Let f ∈ R [ x, y ] be a polynomial with non-singular Hessian curve and support in∆. If f t is the patchworking polynomial of f induced by λ , then there exists δ > < | t | < δ , there is an inclusion ϕ t : T SP P ( f, τ ) ∗ (cid:44) → T SP P ( f t ) ∗ . Here
T SP P ( f, τ ) ∗ := (cid:91) E ∈ τ { ( E, p ); p ∈ T SSP ( f | E ) ∗ } , where for E ∈ τ , T SP P ( f | E ) ∗ denotes the set of transversal special parabolic points in thegraph of the restriction map f | E that lie in π − ( R ∗ ).We use this theorem to disprove a conjecture that first appeared in 2002 in Ortiz’s Phd disserta-tion [11], which was written under Arnold’s supervision. In [6], Hern´andez-Mart´ınez, A. Ortiz.and F. S´anchez-Bringas, give a degree 4 polynomial with 2 special parabolic points above thebound of d ( d −
2) given by A. Ortiz. Since d ( d −
2) + 2 ≤ ( d − d −
9) if d ≥
13, our theorembring us closer to the bound ( d − d −
12) special parabolic points given in [5] in this case.In Section 2, we give preliminaries on the classification of points in a surface and recall knownresults to characterise special parabolic points in the graph of a function f as the zero set ofthree polynomials H f , E ,f and E ,f defined in terms of f . In Section 3, we recall knownresults on convex triangulations and describe Viro’s patchworking technique. In Sections 4,and 5, we describe how the variety defined by H f , E ,f and E ,f behaves under the one-parameter perturbation f + t g t given in terms of polynomials g t ∈ R [ t ][ x, y ]; and in Sections6 and 7, we analyse how the number of transversal special parabolic points is preserved underquasihomotheties of the form ( x, y ) (cid:55)→ ( t α x, t β y ) for t (cid:54) = 0.In Section 8, we state our main result to describe the behaviour of transversal special parabolicpoints under Viro’s patchworking. Lastly, in Section 9, we use Corollary 8.7 to build a one-parameter family f t ∈ R [ t ][ x, y ] of polynomials of degree d with at least ( d − d −
9) transversalspecial parabolic points in their graphs for sufficiently small values of t .The authors would like to thank Erwan Brugall´e and Adriana Ortiz-Rodr´ıguez for their seminarsand for valuable discussions on the subject of real surfaces. In particular, to A. Ortiz-Rodr´ıguezfor the proof of Proposition 2.6. The second author would like to thank Luc´ıa L´opez de Medrano,for answering several questions on the subject.2 Classification of points in a surface
Definition 2.1.
Let S ⊂ R be a surface defined by the vanishing set of a differentiable function F : R → R , that is, S = { ( x, y, z ) ∈ R ; F ( x, y, z ) = 0 } . Take p ∈ S and let l : R → R be thelinear parametrisation of a line with p = l ( t ). The line l has contact order k ∈ N with S at p if and only if the partial derivatives satisfy( F ◦ l ) ( m ) ( t ) = 0 , for m = 0 , . . . , k −
1; and ( F ◦ l ) ( k ) ( t ) (cid:54) = 0 . Tangent lines to a point in a regular surface have contact order k ≥
2. Salmon G. [14] usedthis property to classify the points in a surface according to the following criteria.
Definition 2.2.
Let p be a point in the regular surface S ⊂ R . A line with contact order k ≥ p ∈ S is called an asymptotic direction . A point p ∈ S is called1) elliptic if all tangent lines to S at p have contact order equal to two;2) hyperbolic if it has exactly two asymptotic directions; or3) parabolic if it has either one or more than two asymptotic directions. A parabolic point p is also calleda) generic if it has only one asymptotic direction l and the contact order of l at p is 3;b) special if it has only one asymptotic direction l with contact order k ≥
4; orc) degenerate if it has more than two asymptotic directions.The set of parabolic points in a non-degenerate surface S ⊂ R forms a curve called the paraboliccurve of S .Let S ⊂ R be locally expressed as the graphΓ f = { ( x, y, z ) ∈ R | f ( x, y ) = z } of a differentiable function f : R → R . We will consider from now on the standard projection π : R → R on the xy -plane and we will denote by SP P ( f ) the set of special parabolic pointsin Γ f , and by SP P ( f ) ∗ the points in SP P ( f ) that lie in π − ( R ∗ ).Hereafter, we will denote the vanishing set of a function f as V ( f ). Definition 2.3.
Let f : R → R be a differentiable function . We will refer to the curve V ( H f ),defined by the Hessian H f ( x, y ) := f xx f yy − f xy of f , as the Hessian curve of f .Note that the Hessian H f of f is the determinant of its Hessian matrix Hess ( f ) = (cid:18) f xx f xy f yx f yy (cid:19) .When the graph of a function is a non-degenerate surface, the Hessian of the function, alongwith the following three functions, plays an important role in finding special parabolic points. Definition 2.4.
Let f : R → R be a differentiable function . We consider the functions C f , E ,f and E ,f given byi) C f ( x, y ) := (cid:0) − ( H f ) y ( H f ) x (cid:1) (cid:18) f xx f xy f yx f yy (cid:19) (cid:18) − ( H f ) y ( H f ) x (cid:19) ; andii) (cid:18) E ,f E ,f (cid:19) = (cid:18) f xx f xy f yx f yy (cid:19) (cid:18) − ( H f ) y ( H f ) x (cid:19) , 3here ( H f ) x and ( H f ) y are the partial derivatives of the Hessian of f with respect to x and y ,respectively.Note that C f ( x, y ) = Q f ( − ( H f ) y , ( H f ) x ) , (1)where Q f is the quadratic form Q f ( x, y ) = (cid:0) dx dy (cid:1) (cid:18) f xx f xy f yx f yy (cid:19) (cid:18) dxdy (cid:19) , while the polynomials E ,f and E ,f were introduced by V. I. Arnold in [1]. Proposition 2.5.
Let S ⊂ R be locally expressed as the graph Γ f of a differentiable function f : R → R and let H f be the Hessian of f .i) The projection on the xy -plane of the parabolic curve of Γ f is the Hessian curve of f . Atangent vector to the parabolic curve of Γ f at a point p projects to a vector which is amultiple of the vector ( − ( H f ) y ( π ( p )) , ( H f ) x ( π ( p ))) ∈ R , tangent to the Hessian curve of f at q .ii) Let l : R → R , t (cid:55)→ p + tu with u ∈ R , parametrise a line with contact order k ≥ p ∈ Γ f . Then l is an asymptotic direction of Γ f if and only if the projection π ( u ) is azero of the quadratic form Q f .iii) If the Hessian curve of f is non-singular, then the set of special parabolic points in Γ f isdefined by the intersection of the tangent curves V ( H f ) and V ( C f ).The proof of i) and ii) are straight forward and part iii) is given in [5].Our next result allows us to find special parabolic points in the graph of a differentiable functionin terms of its Hessian curve and the curves V ( E ,f ) and V ( E ,f ). Proposition 2.6.
Let p = ( q, f ( q )) ∈ R be in the graph Γ f of a differentiable function f : R → R . If the Hessian curve of f is non-singular at q , then p ∈ Γ f is an special parabolic pointif and only if q lies in the intersection of the curves V ( H f ), V ( E ,f ) and V ( E ,f ). Proof.
We will prove the forward implication. From Proposition 2.5, p = ( q, f ( q )) ∈ Γ f is a spe-cial parabolic point if and only if q ∈ V ( H f ) ∩ V ( C f ). The condition Q f ( − ( H f ) y ( q ) , ( H f ) x ( q )) = C f ( q ) = 0 given by (1) implies, following Prop. 2.5 ii), that the line l in the tangent plane to Γ f passing through p ∈ Γ f in the direction u ∈ R , with (0 , (cid:54) = π ( u ) = ( − ( H f ) y ( q ) , ( H f ) x ( q )) ∈ R , is an asymptotic direction of Γ f at p .The vector π ( u ) = ( − ( H f ) y ( q ) , ( H f ) x ( q )) is the only zero in R \{ (0 , } of the normal curvaturefunction v (cid:55)→ C f ( v ). Since R \ { (0 , } is homotopically equivalent to S , then π ( u ) is either amaximum or a minimum of the normal curvature function and, thus, π ( u ) is the only eigenvectorof the Hessian matrix of f . Let λ be the eigenvalue associated to π ( u ), then we have 0 = π ( u ) (cid:18) f xx f xy f yx f yy (cid:19) π ( u ) t = λ (cid:107) π ( u ) (cid:107) . Since π ( u ) (cid:54) = (0 , E ,f ( q ) E ,f ( q ) (cid:19) = (cid:18) f xx f xy f yx f yy (cid:19) (cid:18) − ( H f ) y ( q )( H f ) x ( q ) (cid:19) = (cid:18) (cid:19) . The backward implication is straightforward from Definition 2.4.
Corollary 2.7.
Let f be a differentiable function f : R → R and let Γ f be the graph of f . Ifthe Hessian curve of f is non-singular, then SP P ( f ) = { p ∈ Γ f ; π ( p ) ∈ V ( H f ) ∩ V ( E ,f ) ∩ V ( E ,f ) } . Proof.
It follows from Proposition 2.6.
In this section we will recall Viro’s patchworking technique. This procedure was introducedin the late seventies as a technique to glue simple algebraic curves in order to construct realalgebraic non-singular curves with prescribed topology.Let f ∈ R [ x, y ] be a polynomial, the support of f is the finite set of pairs ( i, j ) ∈ Z whoseentries are the exponent of a monomial in f . That is, given f ( x, y ) := (cid:88) a i,j x i y j ,Supp ( f ) := (cid:8) ( i, j ) ∈ Z ; a i,j (cid:54) = 0 (cid:9) . For any subset A ⊂ R , we define the restriction of f to A by f | A ( x, y ) := (cid:88) ( i,j ) ∈ A ∩ Supp ( f ) a i,j x i y j . Let ∆ ⊂ R be a polyhedron and let τ be a polyhedral subdivision of ∆. We say that τ is convex if there exists a convex piecewise linear function λ : ∆ → R ≥ , taking integer values onthe vertices of the subdivision τ , whose restriction to the polyhedra of τ is linear; and with theproperty that it is not linear in the union of any two distinct polyhedra of τ . We will say inthis case that λ induces the convex polyhedral subdivision τ .Given a convex polyhedral subdivision τ induced by the function λ , the graph Γ λ forms apolytope called the compact polytope with polyhedral subdivision induced by λ . We will refer tothe set of 2-dimensional faces that lie in Γ λ by T ( λ ).The projection π : R → R on the xy -plane induces a bijection between the faces of T ( λ ) andthe polyhedra in τ . The inverse of this bijection will be denoted by µ , that is, µ : τ → T ( λ ) , E (cid:55)→ π − ( E ) ∩ Γ λ . Let ∆ ⊂ R be a polyhedron and let λ : ∆ → R ≥ be a convex linear function inducing τ ,a convex polyhedral subdivision of ∆. Let f ( x, y ) = (cid:88) ( i,j ) ∈ ∆ a i,j x i y j ∈ R [ x, y ] be a polynomial5hose support is contained in ∆. The polynomial f t ( x, y ) := (cid:88) ( i,j ) ∈ ∆ a i,j t λ ( i,j ) x i y j ∈ R [ t ][ x, y ],will be called the patchworking polynomial of f induced by λ . Given (cid:101) S ⊂ R , the restriction of f t to (cid:101) S is given by f t | (cid:101) S := (cid:88) ( i,j,λ ( i,j )) ∈ (cid:101) S a ij t λ ( i,j ) x i y j . Given (cid:101) E ∈ T ( λ ) and E ∈ τ such that π ( (cid:101) E ) = E , then f t | (cid:101) E is the patchworking of f E inducedby λ | E . Definition 3.1.
Let ∆ ⊂ R be a polyhedron and let τ be a convex polyhedral subdivisionof ∆ induced by λ : ∆ → R ≥ . Let f ( x, y ) = (cid:88) ( i,j ) ∈ ∆ a i,j x i y j ∈ R [ x, y ] be a polynomial withsupport contained in ∆ and let f t be the patchworking polynomial of f induced by λ . Given r ∈ R ≥ , we will denote by f [ r ] t the restriction of f t to λ − ( r ), i.e., f [ r ] t ( x, y ) := t r (cid:88) { ( i,j ) ∈ ∆; λ ( i,j )= r } a i,j x i y j . Set r := min ( i,j ) ∈ Supp ( f ) λ ( i, j ), then (cid:101) E := { ( a, b, c ) ∈ Γ λ ; c = r } is a face of T ( λ ) and f t | (cid:101) E = f [ r ] t = t r f | π ( (cid:101) E ) = t r f | E , where E = π ( (cid:101) E ) . (2)Let ∆ ⊂ R be a polyhedron with vertices in the integer lattice Z and let f ∈ R [ x, y ] be apolynomial with support in ∆. Let τ be the convex polyhedral subdivision of ∆ induced by λ : ∆ → R ≥ . Denote by CC ( f ) ∗ the set of compact connected components of V ( f ) \ { xy = 0 } and define CC ( f, τ ) ∗ := (cid:91) E ∈ τ { ( E, C ); C ∈ CC ( f | E ) ∗ } . Viro’s construction implies that, under some generic conditions, if f t ∈ R [ t ][ x, y ] is the patch-working polynomial of f induced by λ , then there exists δ > CC ( f, τ ) ∗ (cid:44) → CC ( f t ) ∗ for 0 < | t | < δ . The main purpose of this article is to extend this result to special parabolicpoints. Example 1.
Set f ( x, y ) := x y (1 + x + y + y ), let ∆ := Conv ( { (2 , , (3 , , (2 , , (2 , } ) bethe Newton polyhedron associated to f , and let λ : ∆ → R ≥ be the convex function definedas follows λ ( i, j ) = if i + j ≤ ,i + j − if i + j > . The subdivision of ∆ induced by λ , is τ := { ∆ , ∆ } where ∆ := Conv ( { (2 , , (3 , , (2 , } )and ∆ := Conv ( { (3 , , (2 , , (2 , } ). The patchworking polynomial of f induced by λ is f t = x y (1 + x + y + ty ). For 0 < | t | < .
3, we have the following pictures.6 - - - - - - - - - - - a) - - - - - - b) - - - - - - c)Figure 1: Figure a) shows E f | ∆1 , E f | ∆1 and H f | ∆1 ; figure b) shows E f | ∆2 , E f | ∆2 and H f | ∆2 ;and figure c) shows E f t | ∆ , E f t | ∆ and H f t | ∆ . In this section we recall some definitions and statements on perturbation theory of polynomialsand curves. These statements are consequences of general results in differential topology (seefor example [7]) and are closely related to the concept of transversality. Detailed proofs arealso written in [4].Two non-empty curves C , C ⊂ R intersect transversally at q ∈ C ∩ C , denoted by C (cid:116) q C ,if they are non-singular at q and their tangent lines at q are transversal. Definition 4.1.
Let p = ( q, f ( q )) ∈ R be a special parabolic point of the graph of f ∈ R [ x, y ].We will say that p is a transversal special parabolic point of f if the curve V ( H f ) is non-singularat q ∈ R and V ( E ,f ) (cid:116) q V ( E ,f ). We will denote by T SP P ( f ) the set of transversal specialparabolic points of Γ f .For i = 1 , x, y ) ∈ R in the curve V ( E i,f ) is given byker dE i,f | ( x,y ) . A smooth point p = ( q, f ( q )) ∈ R is a transversal special parabolic point in thegraph of f , if R (cid:39) ker dE ,f | q + ker dE ,f | q . (3)Platonova’s genericity condition [13] implies that special parabolic points are generically trans-versal.Let f ∈ R [ x, y ] be a polynomial. A family of functions F t = f + tg t , where g t ∈ R [ t ][ x, y ], willbe called a perturbation of f . Let C := V ( f ) be the curve defined by the zero set of f ∈ R [ x, y ],the family of curves C t := V ( F t ), defined by a perturbation of f will be called a perturbationof C .Given a point q ∈ R , we will denote by D ( q, r ), the closed disc of radious r centered at q . Proposition 4.2.
Let F t ∈ R [ t ][ x, y ] be a perturbation of f ∈ R [ x, y ]. For δ >
0, let { q t } t ∈ ( − δ,δ ) be a collection of points in D ( q, r ) ⊂ R such that lim t → q t = q , then lim t → F t ( q t ) = f ( q ).7 roposition 4.3. Let C t , D t ⊂ R be perturbations of the curves C, D ⊂ R . If C (cid:116) q D ,then for any r > δ > | t | < δ , the curves C t and D t intersecttransversally at some point q t ∈ D ( q, r ). Moreover, q t can be chosen so that lim t → q t = q .The condition of transversality is crucial in Proposition 4.3. Example 2.
Let C t := V ( y − x − t ) , D t := V ( y + x + t ) be perturbations of the curves C := V ( y − x ) , D := V ( y + x ). The curves C and D have one intersection point, but C ∩ D ,is non-transversal. For positive values of t the intersection C t ∩ D t inside D (0 , r ) is empty,while for negative values of t the intersection C t ∩ D t inside D (0 , r ) has two points, so thenumber of points in the intersection of C and D is not preserved under small perturbations.Proposition 4.3 cannot be extended to more than two curves. Example 3.
Let C t := V ( x − t ) , D t := V ( y − t ) and E t := V ( y + x + t ) be perturbations ofthe curves C := V ( x ) , D := V ( y ) and E := V ( y + x ). The curves C, D and E have only onetransversal intersection point. For small values of | t | the intersection C t ∩ D t ∩ E t inside D (0 , r )is empty, so the number of points in the intersection of C, D and E is not preserved under smallperturbations. Proposition 4.4.
Let C t := V ( F t ) ⊂ R be a perturbation of the curve C := V ( f ) ⊂ R . Let C be non-singular inside D ( q, R ) for q ∈ C . Then there exists δ > | t | < δ , theintersection C t ∩ D ( q, R ) is non-empty and non-singular.Given a non-empty subset A ⊂ R and ε >
0, we will denote by Tub ε ( A ) ⊂ R , the set ofpoints whose distance to A is no greater than ε and call it the tubular neighbourhood of radious ε centered along A , that is, Tub ε ( A ) := (cid:91) q ∈ A D ( q, ε ) . We will denote by Int Tub ε ( A ), the interior of the tubular neighbourhood Tub ε ( A ). Proposition 4.5.
Let C t := V ( F t ) ⊂ R be a perturbation of the curve C := V ( f ) ⊂ R .Given ε > R > q ∈ C , there exists δ > | t | < δ , the intersection C t ∩ D ( q, R ) is contained in the tubular neighbourhood Tub ε ( C ). Special parabolic points can be determined by the intersection of two tangent curves (seeProposition 2.5 iii)), or the intersection of three curves (see Proposition 2.6); however, as wehave seen in examples 2 and 3, both of these situations are generally not preserved under smallperturbations.In [9], E. Landis states that, under some general conditions, special parabolic points are pre-served under perturbations. He doesn’t give a proof of this fact. We gather that this fact is a8onsequence of Platonova’s work. However, here we give a detailed proof for transversal specialparabolic points.
Proposition 5.1.
Let f ∈ R [ x, y ] be a polynomial of degree d ≥
3. If V ( F t ) ⊂ R is aperturbation of the curve V ( f ), then the curves defined by the polynomials H F t , E ,F t and E ,F t , are perturbations of the curves defined by H f , E ,f and E ,f , respectively. Proof.
The Hessian of F t ( x, y ) = f ( x, y ) + t g t ( x, y ) is given by H F t ( x, y ) = H f ( x, y ) + t (cid:101) h t ( x, y ),where (cid:101) h t := ( f xx g tyy + g txx f yy − f xy g txy ) + tH g t and H f , H g t are the Hessians of f and g t ,respectively. Hence, the curve V ( H F t ) is a perturbation of the curve V ( H f ).By definition, the polynomials E ,F t and E ,F t , are given by E ,F t ( x, y ) : = ( − H F t ) y ( F t ) xx + ( H F t ) x ( F t ) xy = E ,f ( x, y ) + t (cid:101) e t ( x, y ) and E ,F t ( x, y ) : = ( − H F t ) y ( F t ) xy + ( H F t ) x ( F t ) yy = E ,f ( x, y ) + t (cid:101) e t ( x, y ) , where (cid:101) e t ( x, y ) = ψ + t ( ψ + tE ,g t ) and (cid:101) e t ( x, y ) = ξ + t ( ξ + tE ,g t ) are perturbations of ψ = − ( H f ) y g txx + ( H f ) x g txy − ϕ y f xx + ϕ x f xy and ξ = − ( H f ) y g txy + ( H f ) x g tyy − ϕ y f xy + ϕ x f yy ,respectively. The remaining polynomials are given by ψ = − ϕ y g txx + ϕ x g txy − ( H g t ) y f xx +( H g ) x f xy and ξ = − ϕ y g txy + ϕ x g tyy − ( H g t ) y f xy + ( H g t ) x f yy , which are given in terms of theHessians of f and g t . This way, the curves V ( E ,F t ) and V ( E ,F t ) are perturbations of thecurves V ( E ,f ) and V ( E ,f ), respectively, as claimed. Corollary 5.2.
Let f ∈ R [ x, y ] be a polynomial of degree d ≥ ⊂ R be a boundedregion in R . If F t is a perturbation of f and the curve V ( H f ) is non-singular inside the closureof Ω, then the Hessian curve V ( H F t ) of F t is non-singular in Ω for sufficiently small values of t . Proof.
Direct consequence of Propositions 4.4 and 5.1.
Lemma 5.3.
Let q be a point in V ( E ,f ) ∩ V ( E ,f ) and let k = H f ( q ). If k (cid:54) = 0, then q isa singular point of the level set curve H − f ( k ) = { ( x, y ) ∈ R ; H f ( x, y ) = k } defined by theHessian of f . Proof.
Take q ∈ V ( E ,f ) ∩ V ( E ,f ) and suppose that q ∈ H − f ( k ) with k (cid:54) = 0. Since (cid:18) (cid:19) = (cid:18) E ,f ( q ) E ,f ( q ) (cid:19) = (cid:18) f xx ( q ) f xy ( q ) f yx ( q ) f yy ( q ) (cid:19) (cid:18) − ( H f ) y ( q )( H f ) x ( q ) (cid:19) , the vector υ ( q ) := ( − ( H f ) y ( q ) , ( H f ) x ( q )) t ∈ ker (cid:18) f xx ( q ) f xy ( q ) f yx ( q ) f yy ( q ) (cid:19) ∩ T q H − f ( k ), where T q H − f ( k )is the tangent space to H − f ( k ) at q . If υ ( q ) (cid:54) = (0 , (cid:18) f xx ( q ) f xy ( q ) f yx ( q ) f yy ( q ) (cid:19) ≤ q ∈ V ( H f ) and we reach a contradiction. Hence υ ( q ) = (0 , q is a singularpoint of H − f ( k ).Our next theorem allows us to relate the transversal special parabolic points in the graph of afunction to those in the graph of any of its perturbations.9 heorem 5.4. Let f ∈ R [ x, y ] be a polynomial of degree d ≥ F t be a perturbation of f . Then, for every ( q, f ( q )) ∈ T SP P ( f ) and ∀ ε > δ > | t | < δ there is a point ( q t , F t ( q t )) ∈ T SP P ( F t ) with q t insidethe closed disc D ( q, ε ) ⊂ R of radious ε . Proof.
Let p = ( q, f ( q )) ∈ R be a transversal special parabolic point in the graph of f and let π : R → R be the projection on the xy -plane.Since p is a transversal special parabolic point, then V ( H f ) is non-singular at q . By Corollary5.2, the Hessian curve of F t is non-singular inside D ( q, ε ) for small values of t , and by Proposition2.6 there exists δ > | t | < δ , π ( T SP P ( F t )) ∩ D ( q, ε ) = V ( H F t ) ∩ V ( E ,F t ) ∩ V ( E ,F t ) ∩ D ( q, ε ) . By Proposition 4.3, there exists δ > | t | < δ the curves V ( E ,F t ) and V ( E ,F t )intersect transversally at some q t ∈ D ( q, ε ) with lim t → q t = q .We claim that ( q t , F t ( q t )) ∈ T SP P ( F t ). To prove our claim it is enough to show that q t ∈ V ( H F t ). Suppose that H F t ( q ) = k t (cid:54) = 0, by Lemma 5.3, q t ∈ D ( q, ε ) is a singular point of V k t := { ( x, y ) ∈ R ; H F t ( x, y ) = k t } . The vector υ ( q t ) := ( − ( H F t ) y ( q t ) , ( H F t ) x ( q t )) = (0 , , (cid:54) = ( − ( H f ) y ( q ) , ( H f ) x ( q )) = lim t → ( − ( H f ) y ( q t ) , ( H f ) x ( q t )) P rop. . = lim t → υ ( q t ) = (0 , . Therefore for | t | < min { δ , δ } , the point q t ∈ V ( E ,F t ) ∩ V ( E ,F t ) lies also in V ( H F t ) ∩ D ( q, ε )and ( q t , F t ( q t )) ∈ SP P ( F t ) is, henceforth, a transversal special parabolic point in the graph of F t with q t ∈ D ( q, ε ). Corollary 5.5.
Let f ∈ R [ x, y ] be a polynomial of degree d ≥ F t ∈ R [ t ][ x, y ] be a perturbation of f . Then, for ε > δ > < | t | < δ there exists an inclusion ψ t : T SP P ( f ) (cid:44) → T SP P ( F t ) (4)such that π ( ψ t ( p )) ∈ D ( π ( p ) , ε ). Furthermore, choosing ε small enough, we also have ψ t : T SP P ( f ) ∗ (cid:44) → T SP P ( F t ) ∗ . (5) Proof.
The set
T SP P ( f ) is finite. Let 0 < ε < ε be such that for any q, q (cid:48) ∈ π ( T SP P ( f ))with q (cid:54) = q (cid:48) , D ( q, ε ) ∩ D ( q (cid:48) , ε ) = ∅ and D ( q, ε ) , D ( q (cid:48) , ε ) ⊂ ( R ∗ ) . By Theorem 5.4, for any q ∈ π ( T SP P ( f )), there exists δ q > | t | < δ q , there is a point ( q t , F t ( q t )) ∈ T SP P ( F t ) with q t ∈ D ( q, ε ).Let δ := min q ∈ π ( T SP P ( f )) δ q > p = ( q, f ( q )) ∈ T SP P ( f ) define ψ t ( p ) := ( q t , F t ( q t )).10 On quasihomothetic maps
In this section, we show some properties of a special type of transformations, called quasiho-motheties by O. Y. Viro [15]. Quasihomotheties are maps ρ ( α,β ) s : R → R ( x, y ) (cid:55)→ ( s α x, s β y ) . for some α, β ∈ Z and s ∈ R . For s (cid:54) = 0 the function ρ ( α,β ) s is one-to-one, and the differential dρ ( α,β ) s | ( x,y ) corresponds to the isomorphism defined by the matrix (cid:18) s α s β (cid:19) . Lemma 6.1.
Let h s αi : R → R , x (cid:55)→ s α i x . Given α < α , for any two intervals [ a, b ] , [ c, d ] ⊂ R \ { } , there exists δ > < | s | < δ we have h s α ([ a, b ]) ∩ h s α ([ c, d ]) = ∅ . Proof.
We will give the proof in the case where [ a, b ] , [ c, d ] ⊂ R > . Consider δ = ( ad ) α − α .If s > α , α are even, then h s α ([ a, b ]) = [ s α a, s α b ] and h s α ([ c, d ]) = [ s α c, s α d ], thuschoosing 0 < | s | < δ gives s α − α < ad , so s α d < s α a and h s α ([ a, b ]) ∩ h s α ([ c, d ]) = ∅ . If s < α + α is odd, then either h s α ([ a, b ]) ⊂ R > and h s α ([ c, d ]) ⊂ R < ; or h s α ([ a, b ]) ⊂ R < and h s α ([ c, d ]) ⊂ R > , so the result follows. If s < α , α are odd, then h s α ([ a, b ]) =[ s α b, s α a ] and h s α ([ c, d ]) = [ s α d, s α c ], hence | s | < δ gives s > − ( ad ) α − α , thus 0 < − s < ( ad ) α − α , and s α a < s α d , so h s α ([ a, b ]) ∩ h s α ([ c, d ]) = ∅ .For the cases where [ a, b ] ⊂ R > and [ c, d ] ⊂ R < ; [ a, b ] ⊂ R < and [ c, d ] ⊂ R > ; or [ a, b ] , [ c, d ] ⊂ R < , it is enough to consider 0 < | s | smaller than δ = | ac | α − α , δ = | bd | α − α , δ = ( bc ) α − α to have h s α ([ a, b ]) ∩ h s α ([ c, d ]) = ∅ , respectively. Taking δ := min { δ , δ , δ , δ } , the resultfollows. Proposition 6.2.
Given ( α, β ) (cid:54) = ( α (cid:48) , β (cid:48) ) ∈ Z , let A, B ⊂ ( R ∗ ) be finite sets of points andlet ε > q ∈ A ∪ B , the closed disc D ( q, ε ) ⊂ ( R ∗ ) . Then, there exists δ > < | s | < δ , ρ ( α,β ) s ( T ub ε ( A )) (cid:92) ρ ( α (cid:48) ,β (cid:48) ) s ( T ub ε ( B )) = ∅ . Proof.
Suppose that α (cid:54) = α (cid:48) and let π : R → R be the projection ( x, y ) (cid:55)→ x . Let [ a, b ] , [ e, f ] ⊂ R < , and [ c, d ] , [ g, h ] ⊂ R > be intervals such that π ( T ub ε ( A )) ⊆ [ a, b ] ∪ [ c, d ] and π ( T ub ε ( B )) ⊆ [ e, f ] ∪ [ g, h ] . Then, π (cid:0) ρ ( α,β ) s ( T ub ε ( A )) (cid:1) = { s α x ; x ∈ π ( T ub ε ( A )) } ⊆ h s α ([ a, b ]) ∪ h s α ([ c, d ]) and π (cid:16) ρ ( α (cid:48) ,β (cid:48) ) s ( T ub ε ( B )) (cid:17) = (cid:110) s α (cid:48) x ; x ∈ π ( T ub ε ( B )) (cid:111) ⊆ h s α ([ e, f ]) ∪ h s α ([ g, h ]) . Taking these intervals two-to-two, the result follows from Lemma 6.1.If, on the other hand, α = α (cid:48) and β (cid:54) = β (cid:48) , then with the projection π : R → R , ( x, y ) (cid:55)→ y anda similar process we obtain the result wanted.11 Special parabolic points under quasihomothetic maps
In this section we show how the number of transversal special parabolic points is preservedunder quasihomotheties.Given α, β, r ∈ Z , we will consider the transformation (cid:101) h ( α,β,r ) : R [ x, y ] → R [ s ][ x, y ] (6) f ( x, y ) (cid:55)→ (cid:101) h ( α,β,r ) ( f )( x, y ) = s r f ◦ ρ ( α,β ) s ( x, y ) . Note that (cid:101) h ( α,β,r ) ( f ) does not define a perturbation of f ∈ R [ x, y ]. However, if we considerthe translation T : R [ s ][ x, y ] → R [ s ][ x, y ], f ( x, y, s ) (cid:55)→ f ( x, y, s + 1), then the composition T ◦ (cid:101) h ( α,β,r ) ( f ) is a perturbation of the polynomial f . Lemma 7.1.
Let Ω ⊂ R be a bounded region. If the Hessian curve of f ∈ R [ x, y ] is non-singular inside the closure of Ω, then for s (cid:54) = 0 and r ∈ Z , i) ( x, y, f ( x, y )) ∈ SP P ( f ) ∩ π − (Ω) if and only if ( x, y, (cid:101) h (0 , ,r ) ( f )( x, y )) ∈ SP P ( (cid:101) h (0 , ,r ) ( f )) ∩ π − (Ω).ii) ( x, y, f ( x, y )) ∈ SP P ( f ) ∗ ∩ π − (Ω) if and only if ( x, y, (cid:101) h (0 , ,r ) ( f )( x, y )) ∈ SP P ( (cid:101) h (0 , ,r ) ( f )) ∗ ∩ π − (Ω). Proof.
Set (cid:101) f ( x, y ) := (cid:101) h (0 , ,r ) ( f )( x, y ) = s r f ( x, y ), and let π : R → R be the projection on the xy -plane, by Proposition 2.6 the set of special parabolic points in the graph of (cid:101) f is given by theset SP P ( (cid:101) f ) ∩ π − (Ω) = { ( x, y, (cid:101) f ( x, y )); ( x, y ) ∈ V ( H (cid:101) f ) ∩ V ( E , (cid:101) f ) ∩ V ( E , (cid:101) f ) } ∩ π − (Ω), where H (cid:101) f ( x, y ) = det (cid:32) (cid:101) f xx ( x, y ) (cid:101) f xy ( x, y ) (cid:101) f yx ( x, y ) (cid:101) f yy ( x, y ) (cid:33) = det (cid:18) s r f xx ( x, y ) s r f xy ( x, y ) s r f yx ( x, y ) s r f yy ( x, y ) (cid:19) = s r H f ( x, y ) , and (7) (cid:18) E , (cid:101) f ( x, y ) E , (cid:101) f ( x, y ) (cid:19) = (cid:32) (cid:101) f xx ( x, y ) (cid:101) f xy ( x, y ) (cid:101) f yx ( x, y ) (cid:101) f yy ( x, y ) (cid:33) (cid:18) − ( H (cid:101) f ) y ( x, y )( H (cid:101) f ) x ( x, y ) (cid:19) = (cid:18) s r f xx ( x, y ) s r f xy ( x, y ) s r f yx ( x, y ) s r f yy ( x, y ) (cid:19) (cid:18) − s r ( H f ) y ( x, y ) s r ( H f ) x ( x, y ) (cid:19) = s r (cid:18) E ,f ( x, y ) E ,f ( x, y ) (cid:19) . (8)Since the Hessian curve of f is non-singular inside Ω, by equation (7) also the Hessian curveof (cid:101) f is non-singular inside Ω. Since we have SP P ( f ) ∩ π − (Ω) = { ( x, y, f ( x, y )); ( x, y ) ∈ V ( H f ) ∩ V ( E ,f ) ∩ V ( E ,f ) }∩ π − (Ω), equations (7) and (8) give the results wanted for s (cid:54) = 0. Lemma 7.2.
Let Ω ⊂ R be a bounded region. If the Hessian curve of f ∈ R [ x, y ] is non-singular inside the closure of Ω, then for s (cid:54) = 0 and r ∈ Z , i) ( x, y, f ( x, y )) ∈ T SP P ( f ) ∩ π − (Ω) if and only if ( x, y, (cid:101) h (0 , ,r ) ( f )( x, y )) ∈ T SP P ( (cid:101) h (0 , ,r ) ( f )) ∩ π − (Ω).ii) ( x, y, f ( x, y )) ∈ T SP P ( f ) ∗ ∩ π − (Ω) if and only if ( x, y, (cid:101) h (0 , ,r ) ( f )( x, y )) ∈ T SP P ( (cid:101) h (0 , ,r ) ( f )) ∗ ∩ π − (Ω). Proof.
A smooth point p = ( q, f ( q )) ∈ R is a transversal special parabolic point in the graph of f if it satisfies equation (3). By equations (7) and (8) we have that R (cid:39) ker dE , (cid:101) f | q +ker dE , (cid:101) f | q ,and by Lemma 7.1 we have the result. 12et (cid:101) ϕ ( α,β,r ) s to be the transformation given by (cid:101) ϕ ( α,β,r ) s : Γ f → Γ (cid:101) h ( α,β,r ) ( f ) (9)( x, y, f ( x, y )) (cid:55)→ (cid:16) s − α x, s − β y, (cid:101) h ( α,β,r ) ( f ) (cid:0) ρ ( − α, − β ) s ( x, y ) (cid:1)(cid:17) . Proposition 7.3.
Let Ω ⊂ R be a bounded region. If the Hessian curve of f ∈ R [ x, y ] isnon-singular inside the closure of Ω, then for s (cid:54) = 0, and α, β, r ∈ Z , the mapping (cid:101) ϕ ( α,β,r ) s gives aone-to-one correspondence between SP P ( f ) ∩ π − (Ω) and SP P ( (cid:101) h ( α,β,r ) ( f )) ∩ π − ( ρ ( − α, − β ) s (Ω)).Moreover, (cid:101) ϕ ( α,β,r ) s gives a one-to-one correspondence between SP P ( f ) ∗ ∩ π − (Ω) and the set SP P ( (cid:101) h ( α,β,r ) ( f )) ∗ ∩ π − ( ρ ( − α, − β ) s (Ω)). Proof.
By Lemma 7.1, it is enough to prove that (cid:101) ϕ ( α,β,r ) s gives a one-to-one correspondence for r = 0. Set (cid:98) f ( x, y ) := (cid:101) h ( α,β, ( f )( x, y ) = f ( ρ ( α,β ) s ( x, y )). By Proposition 2.6, the set of specialparabolic points in the graph of (cid:98) f is described by the set SP P ( (cid:98) f ) ∩ π − (Ω) = { ( x, y, (cid:98) f ( x, y )); ( x, y ) ∈ V ( H (cid:98) f ) ∩ V ( E , (cid:98) f ) ∩ V ( E , (cid:98) f ) } ∩ π − (Ω) , where H (cid:98) f ( x, y ) = det (cid:32) (cid:98) f xx ( x, y ) (cid:98) f xy ( x, y ) (cid:98) f yx ( x, y ) (cid:98) f yy ( x, y ) (cid:33) = det (cid:32) s α f xx ( ρ ( α,β ) s ( x, y )) s α + β f xy ( ρ ( α,β ) s ( x, y )) s α + β f yx ( ρ ( α,β ) s ( x, y )) s β f yy ( ρ ( α,β ) s ( x, y )) (cid:33) = s α + β ) H f ( ρ ( α,β ) s ( x, y )) , and (10) (cid:18) E , (cid:98) f ( x, y ) E , (cid:98) f ( x, y ) (cid:19) = (cid:32) (cid:98) f xx ( x, y ) (cid:98) f xy ( x, y ) (cid:98) f yx ( x, y ) (cid:98) f yy ( x, y ) (cid:33) (cid:18) − ( H (cid:98) f ) y ( x, y )( H (cid:98) f ) x ( x, y ) (cid:19) = (cid:32) s α f xx ( ρ ( α,β ) s ( x, y )) s α + β f xy ( ρ ( α,β ) s ( x, y )) s α + β f yx ( ρ ( α,β ) s ( x, y )) s β f yy ( ρ ( α,β ) s ( x, y )) (cid:33) (cid:18) − s α + β ) ( H f ) y ( s α x, s β y ) s α + β ) ( H f ) x ( s α x, s β y ) (cid:19) = (cid:32) s α +3 β E ,f ( ρ ( α,β ) s ( x, y )) s α +4 β E ,f ( ρ ( α,β ) s ( x, y )) (cid:33) . (11)By equation (10), the Hessian curve V ( H (cid:98) f ) of (cid:98) f is non-singular inside Ω for s (cid:54) = 0; andby equations (10) and (11), (cid:101) ϕ ( α,β,r ) s is one-to-one between the sets SP P ( f ) ∩ π − (Ω) and SP P ( (cid:101) h ( α,β,r ) ( f )) ∩ π − ( ρ ( − α, − β ) s (Ω)). Moreover, since for s (cid:54) = 0, s α x = 0 if and only if x = 0, and s β y = 0 if and only if y = 0, the latest claim is proved. Proposition 7.4.
Let Ω ⊂ R be a bounded region. If the Hessian curve of f ∈ R [ x, y ] is non-singular inside Ω, then for s (cid:54) = 0 and α, β, r ∈ Z , (cid:101) ϕ ( α,β,r ) s gives a one-to-one correspondence be-tween T SP P ( f ) ∩ π − (Ω) and T SP P ( (cid:101) h ( α,β,r ) ( f )) ∩ π − ( ρ ( − α, − β ) s (Ω)). Moreover, (cid:101) ϕ ( α,β,r ) s is a one-to-one correspondence between T SP P ( f ) ∗ ∩ π − (Ω) and T SP P ( (cid:101) h ( α,β,r ) ( f )) ∗ ∩ π − ( ρ ( − α, − β ) s (Ω)). Proof.
By Corollary 7.2, it is enough to prove the proposition for r = 0. Set (cid:98) f ( x, y ) := (cid:101) h ( α,β, ( f )( x, y ) = f ◦ ρ ( α,β ) s ( x, y ). By Proposition 7.3, it is enough to show that if R (cid:39) dE ,f | ( x,y ) + ker dE ,f | ( x,y ) , then there exists δ > < | s | < δ we have R (cid:39) ker dE , (cid:98) f | ρ ( − α, − β ) s ( x,y ) + ker dE , (cid:98) f | ρ ( − α, − β ) s ( x,y ) . Since dE , (cid:98) f | ( x,y ) = s α +3 β dE ,f | ρ ( α,β ) s ( x,y ) · (cid:18) s α s β (cid:19) , and dE , (cid:98) f | ( x,y ) = s α +4 β dE ,f | ρ ( α,β ) s ( x,y ) · (cid:18) s α s β (cid:19) , then ker dE i, (cid:98) f | ρ ( − α, − β ) s ( x,y ) is the image of ker dE i,f | ( x,y ) under the isomorphism defined by thematrix dρ ( α,β ) s | ( x,y ) = (cid:18) s α s β (cid:19) , for i = 1 ,
2. Therefore, if R (cid:39) ker dE ,f | ( x,y ) + ker dE ,f | ( x,y ) ,then ker dE , (cid:98) f | ρ ( − α, − β ) s ( x,y ) + ker dE , (cid:98) f | ρ ( − α, − β ) s ( x,y ) (cid:39) R .Let f ∈ R [ x, y ] be a polynomial and let f t be a perturbation of f . For t ∈ R , set h ( α,β,r ) to bethe transformation h ( α,β,r ) : R [ t ][ x, y ] → R [ t ][ x, y ] f t ( x, y ) (cid:55)→ h ( α,β,r ) ( f t )( x, y ) = t r f t ◦ ρ ( α,β ) t ( x, y ) . Note that h ( α,β,r ) can be obtained by extending the transformation (cid:101) h ( α,β,r ) in (6) to polynomials f t ∈ R [ t ][ x, y ] and composing with (cid:37) : R [ s, t ][ x, y ] → R [ t ][ x, y ], P ( s, t, x, y ) (cid:55)→ P ( t, t, x, y ).Set ϕ ( α,β,r ) t to be the transformation given by ϕ ( α,β,r ) t : Γ f t → Γ h ( α,β,r ) ( f t ) ( x, y, f t ( x, y )) (cid:55)→ (cid:16) t − α x, t − β y, h ( α,β,r ) ( f t ) ◦ ρ ( − α, − β ) t ( x, y ) (cid:17) . Note that ϕ ( α,β,r ) t can be obtained by extending the transformation (cid:101) ϕ ( α,β,r ) s in (9) to polynomials f t ∈ R [ t ][ x, y ] and composing with (cid:37) : R [ s, t ][ x, y ] → R [ t ][ x, y ], P ( s, t, x, y ) (cid:55)→ P ( t, t, x, y ). Proposition 7.5.
Let f ∈ R [ x, y ] be a polynomial with non-singular Hessian curve and let f t ∈ R [ t ][ x, y ] be a perturbation of f . Then, for any bounded region Ω ⊂ R there exist δ > < | t | < δ , ϕ ( α,β,r ) t is a one-to-one correspondence betweeni) SP P ( f t ) ∩ π − (Ω) and SP P ( h ( α,β,r ) ( f t )) ∩ π − (cid:16) ρ ( − α, − β ) t (Ω) (cid:17) ,ii) SP P ( f t ) ∗ ∩ π − (Ω) and SP P ( h ( α,β,r ) ( f t )) ∗ ∩ π − (cid:16) ρ ( − α, − β ) t (Ω) (cid:17) ,iii) T SP P ( f t ) ∩ π − (Ω) and T SP P ( h ( α,β,r ) ( f t )) ∩ π − (cid:16) ρ ( − α, − β ) t (Ω) (cid:17) , andiv) T SP P ( f t ) ∗ ∩ π − (Ω) and T SP P ( h ( α,β,r ) ( f t )) ∗ ∩ π − (cid:16) ρ ( − α, − β ) t (Ω) (cid:17) for any ( α, β, r ) ∈ Z . Proof.
Since H f is non-singular in the closure of Ω, by Corollary 5.2, there exists δ > < | t | < δ , H f t has no singularities inside Ω. By Proposition 7.3, making s = t , wehave i) and ii). And, by Proposition 7.4, we have iii) and iv).14 Viro’s Theorem for transversal special parabolic points
In this section we will adapt Viro’s patchworking technique to the study of transversal specialparabolic points on the graphs of polynomials.
Proposition 8.1.
Given a convex polyhedral subdivision τ of ∆ ⊂ R induced by λ : ∆ → R ≥ ,there exists d ∈ Z > such that d · λ (∆ ∩ Z ) ⊂ Z . Proof.
Take α ∈ ∆ ∩ Z . If α is a 0-dimensional polyhedron in τ , then λ ( α ) ∈ Z . Otherwise,let F ∈ τ be the polyhedron with vertices V , . . . , V s such that α ∈ F . There exist rationalnumbers α i ∈ Q such that α = s (cid:88) i =1 α i V i and thus λ ( α ) = s (cid:88) i =1 α i λ ( V i ) ∈ Q . The result followsfrom the fact that λ (∆ ∩ Z ) is a finite set. Lemma 8.2.
Let ∆ ⊂ R be a polyhedron and let τ be the polyhedral subdivision of ∆ inducedby the convex function λ : ∆ → R ≥ . If λ (∆ ∩ Z ) ⊂ Z , then for (cid:101) E ∈ T ( λ ) the only vector ofthe form ( α, β, ∈ R that is orthogonal to (cid:101) E has integer coordinates. Proof.
Let ( x , y , λ ( x , y )) , ( x , y , λ ( x , y )) , ( x , y , λ ( x , y )) ∈ (cid:101) E ∩ Z be points with theproperty that there are no points with integer coordinates inside the triangle ∆( v , v , v ) withvertices v = ( x , y ) , v = ( x , y ) , v = ( x , y ) ∈ Z , that is,1 = area (∆( v , v , v )) = | ( v − v ) · ( v − v ) | . (12)A vector ( α, β, ∈ R is orthogonal to (cid:101) E if ( α, β, · ( x k , y k , λ ( x k , y k )) is constant for k = 1 , , α, β and r , α β x k y k λ ( x k , y k ) = x k y k r k = 1 , , , (13)sending (cid:101) E to the horizontal plane { Z = r } is equivalent to the system of three equations x y x y x y αβ − r = − λ ( x , y ) − λ ( x , y ) − λ ( x , y ) . The determinant of the 3 × x y − y x ) − ( x y − x y ) + ( x y − x y ) = ( x − x , y − y ) · ( − ( y − y ) , x − x ) =( v − v ) · ( v − v ) ⊥ , is equal to ± α, β, r ∈ Z tothe system (13) is granted. Lemma 8.3.
Let ∆ ⊂ R be a polyhedron and let τ be the polyhedral subdivision of ∆ inducedby the convex function λ : ∆ → R ≥ . If λ (∆ ∩ Z ) ⊂ Z and ( α, β, ∈ Z is orthogonal to (cid:101) E ∈ T ( λ ), then the linear transformation l ( α,β ) : R → R defined by the matrix α β ,satisfies l ( α,β ) ( (cid:101) E ) ⊂ { Z = r } where r := min { ( α, β, · v ; v ∈ T ( λ ) } is an integer number.15 roof. The linear transformation l ( α,β ) sends the face (cid:101) E ∈ T ( λ ) to a face in T ( (cid:101) λ ) contained inthe horizontal plane { Z = r } , where (cid:101) λ ( i, j ) = λ ( i, j )+ iα + jβ ; while the remaining 2-dimensionalfaces in T ( λ ) are sent to faces in T ( (cid:101) λ ) above this horizontal plane, thus r = min { ( α, β, · v ; v ∈ T ( λ ) } ∈ Z .From now on, and without loss of generality, by Proposition 8.1, all our convex polyhedralsubdivisions will be induced by convex functions sending points with integer coordinates tointeger values.Let τ be a convex polyhedral subdivision of ∆ ⊂ R induced by λ : ∆ → R ≥ . Let f ∈ R [ x, y ]be a polynomial with support in ∆. Given ( α, β ) ∈ Z the mapping (cid:101) λ : ( i, j ) (cid:55)→ λ ( i, j ) + αi + βj is also a convex function inducing τ . Let f t be the patchworking polynomial of f induced by λ and let (cid:101) f t be the patchworking polynomial induced by (cid:101) λ , then (cid:101) f t = h ( α,β, ( f t ) = t r f t ( t α x, t β y ) (14)for some integer value r ∈ Z . Proposition 8.4.
Let τ be a convex polyhedral subdivision of ∆ ⊂ R induced by λ : ∆ → R ≥ .Let f ∈ R [ x, y ] be a polynomial with support in ∆ and let f t be the patchworking polynomial of f induced by λ . If the vector ( α, β, ∈ Z is orthogonal to (cid:101) E ∈ T ( λ ), then the patchworkingpolynomial (cid:101) f t induced by (cid:101) λ ( i, j ) = λ ( i, j ) + iα + jβ satisfies h ( α,β, (cid:0) f t | (cid:101) E (cid:1) = (cid:101) f [ r ] t for someconstant r ∈ Z . Proof.
This result is direct consequence of (14) and Lemma 8.3.
Theorem 8.5.
Let ∆ ⊂ R be a polyhedron with vertices in Z and let τ be a convex polyhedralsubdivision of ∆ induced by λ : ∆ → R ≥ . Let f ∈ R [ x, y ] be a polynomial with support in∆ and let f t be the patchworking polynomial of f induced by λ . If the vector ( α, β, ∈ Z isorthogonal to (cid:101) E ∈ T ( λ ) and γ = − min { ( α, β, · υ ; υ ∈ T ( λ ) } then h ( α,β,γ ) ( f t ) is a perturbationof f | π ( (cid:101) E ) . Proof.
Set f ( x, y ) = (cid:88) ( i,j ) ∈ ∆ a i,j x i y j ∈ R [ x, y ]. By Proposition 8.4, the patchworking polynomial (cid:101) f t induced by (cid:101) λ ( i, j ) = λ ( i, j ) + iα + jβ satisfies h ( α,β, ( f t | (cid:101) E ) = (cid:101) f [ r ] t for r = min { ( α, β, · υ ; υ ∈ T ( λ ) } ∈ Z . The polynomial h ( α,β, − r ) ( f t )( x, y ) = t − r (cid:101) f t ( t α x, t β y ) = (cid:88) ( i,j ) ∈ ∆ a i,j t λ ( i,j )+ iα + jβ − r x i y j ∈ R [ t ][ x, y ] is the patchworking polynomial (cid:98) f t induced by the convex function (cid:98) λ : ∆ → R ≥ , (cid:98) λ ( i, j ) := λ ( i, j ) + iα + jβ − r . In particular, h ( α,β, − r ) ( f t | (cid:101) E ) = (cid:98) f [0] t . Expressing (cid:98) f t as the finitesum of level sets (cid:98) λ − ( r ) , . . . (cid:98) λ − ( r m ) of (cid:98) λ corresponding to the values 0 = r < r < · · · < r m ,we have that (cid:98) f t ( x, y ) = f | π ( (cid:101) E ) + tϕ t , with ϕ t ( x, y ) = m (cid:88) l =1 a i,j t r l − (cid:98) f [ r l ] t ( x, y ) ∈ R [ t ][ x, y ], aswanted. 16et ∆ ⊂ R be a polyhedron with vertices in the integer lattice Z and let f ∈ R [ x, y ] be apolynomial with support in ∆. Let τ be the convex polyhedral subdivision of ∆ induced by λ : ∆ → R ≥ . Denote by T SP P ( f, λ ) ∗ the set of pairs T SP P ( f, τ ) ∗ := (cid:91) E ∈ τ { ( E, p ); p ∈ T SSP ( f | E ) ∗ } . Theorem 8.6. (Viro’s Theorem for transversal special parabolic points) Let ∆ ⊂ R be apolyhedron with vertices in Z and let τ be the convex polyhedral subdivision of ∆ induced by λ : ∆ → R ≥ . Let f ∈ R [ x, y ] be a polynomial with non-singular Hessian curve, and supportin ∆. If f t is the patchworking polynomial of f induced by λ , then there exists δ > < | t | < δ , there is an inclusion ϕ t : T SP P ( f, τ ) ∗ (cid:44) → T SP P ( f t ) ∗ . Proof.
Let ( α, β, ∈ Z be an orthogonal vector to the face (cid:101) E ∈ T ( λ ) and let r := − min { ( α, β, · υ ; υ ∈ T ( λ ) } ∈ Z . Then, by Theorem 8.5, the polynomial h ( α,β,r ) ( f t )( x, y ) defines a perturba-tion of f | E , where E = π ( (cid:101) E ) is in τ .For each face E in τ , set C E := π ( T SP P ( f | E ) ∗ ). Choose ε > q, q (cid:48) ∈ C E , we have D ( q, ε ) , D ( q (cid:48) , ε ) ⊂ ( R ∗ ) and D ( q, ε ) ∩ D ( q (cid:48) , ε ) (cid:54) = ∅ . By Corollary 5.5,there exists δ > < | t | < δ there is an inclusion ψ Et : T SP P ( f | E ) ∗ (cid:44) → T SP P ( h ( α,β,r ) ( f t )) ∗ satisfying π ( ψ Et ( p )) ∈ D ( π ( p ) , ε ). Thus, for each q ∈ C E , π (cid:0) T SP P ( h ( α,β,r ) ( f t ) (cid:1) ∗ ∩ D ( q, ε ) (cid:54) = ∅ .By Proposition 7.5, there exists δ > < | t | < δ , (cid:16) ϕ ( α,β,r ) t (cid:17) − : T SP P ( h ( α,β,r ) ( f t )) ∗ → T SP P ( f t ) ∗ is a bijection. Hence, for 0 < | t | < min { δ , δ } , we have the inclusion ϕ Et : T SP P ( f | E ) ∗ ψ Et (cid:44) → T SP P ( h ( α,β,r ) ( f t )) ∗ ( ϕ ( α,β,r ) t ) − (cid:39) −→ T SP P ( f t ) ∗ . By Proposition 6.2, there exists δ >
0, such that, for | t | < δ , Im( ϕ Et ) ∩ Im( ϕ E (cid:48) t ) = ∅ for E (cid:54) = E (cid:48) . Therefore, for 0 < | t | < δ := min { δ , δ , δ } , we obtain an inclusion ϕ t : T SP P ( f, τ ) ∗ (cid:44) → T SP P ( f t ) ∗ . Corollary 8.7.
Let ∆ ⊂ R be a polyhedron with vertices in Z and let τ be the convexpolyhedral subdivision of ∆ induced by λ : ∆ → R ≥ . Let f ∈ R [ x, y ] be a polynomial withsupport in ∆. If f t is the patchworking polynomial of f induced by λ , then there exists δ > < | t | < δ , we have | T SP P ( f, τ ) ∗ | ≤ | T SP P ( f t ) ∗ | . Proof.
It follows from Theorem 8.6. 17
An Application of Corollary 8.7
In this section, we give an example to show how to use Corollary 8.7 to build families ofpolynomials with a prescribed number of transversal especial parabolic points.Let ∆ { a, b, c } ⊂ R denote the triangle with vertices a , b , c ∈ R . We will denote by • ∆ d := ∆ { (2 , , ( d − , , (2 , d − } , • ∆ i, := ∆ { ( i, , ( i + 1 , , ( i, } , • ∆ k,l ) := ∆ { ( k + 1 , l ) , ( k, l + 1) , ( k, l + 2) } , and • ∆ k,l ) := ∆ { ( k + 1 , l ) , ( k, l + 2) , ( k + 1 , l + 1) } for d, i, k, l ∈ Z ≥ . (cid:54) (cid:45)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65) (2 , (cid:64)(cid:64)(cid:65)(cid:65)(cid:65)(cid:65) Triangle ∆ k,l ) (cid:27)(cid:64)(cid:64)(cid:65)(cid:65)(cid:65)(cid:65) Triangle ∆ k,l ) (cid:27)(cid:64)(cid:64) Triangle ∆ i, (cid:27) The triangular subdivision τ of ∆ d obtained by dividing ∆ d into ∆ i, , ∆ k,l ) , ∆ k,l ) with i ∈{ , , . . . , d } , k ∈ { , , . . . , d − } and l ∈ { , . . . , d − k − } , is convex. This subdivision isinduced by a convex function λ : ∆ d → R ≥ that has been used in several works, for examplein [2] and [10]. Consider the polynomial types: P i, ( x, y ) := x i y (1 + x + y ), P k,l ) ( x, y ) := x k y l ( x + y + y ) and P k,l ) ( x, y ) := x k y l ( x + xy + y ), whose support lies in ∆ i, , ∆ k,l ) and∆ k,l ) , respectively. Theorem 9.1.
Let f = x y g ( x, y ) be the degree d polynomial with support in the triangle ∆ d ,where g ∈ R [ x, y ] is a complete polynomial of degree d −
4. Let τ be the polyhedral subdivisioninduced by λ : ∆ d → R ≥ as above, and let f t ∈ R [ x, y ] be the patchworking polynomial of f induced by λ . Then, for d ≤ , ε > | T SP P ( f t ) ∗ | ≥ ( d − d − < | t | < ε . 18 roof. The convex polytope T ( λ ) is the union of d − (cid:101) ∆ i, and ( d − − ( d − = ( d − d − faces of type (cid:101) ∆ k,l ) and (cid:101) ∆ k,l ) , respectively, whose proyections on the xy -plane are given by π (cid:16) (cid:101) ∆ i, (cid:17) = ∆ i, , π (cid:16) (cid:101) ∆ k,l ) (cid:17) = ∆ k,l ) and π (cid:16) (cid:101) ∆ k,l ) (cid:17) = ∆ k,l ) .The restrictions of f to the faces in T ( λ ) are equal to f | ∆ i, = P i, , f | ∆ k,l ) = P k,l ) and f | ∆ k,l ) = P k,l ) . Using computer software Mathematica [18], we can see that, given d ≤ | T SP P ( P i, ) ∗ | =1 for i ≥ | T SP P ( P k,l ) ) ∗ | = 1 and | T SP P ( P k,l ) ) ∗ | = 3 for k ∈ { , , . . . , d − } and l ∈ { , . . . , d − k − } . Corollary 8.7 guarantees that the number of transversal special parabolicpoints in the graph of f t ∈ R [ t ][ x, y ] satisfies | T SP P ( f t ) ∗ | ≥ d −
4) + 4 (cid:18) ( d − d − (cid:19) = ( d − d − , for sufficiently small values of t (cid:54) = 0.Although we haven’t found a proof of the fact that the inequality | T SP P ( P k,l ) ) ∗ | ≥ k, l ∈ Z ≥ , we firmly believe that the bound we give in the statement of Theorem 9.1 worksin general so the restriction on the degree can be removed from its hypotheses. References [1] V. I. Arnold. Remarks on the parabolic curves on surfaces and on the higher-dimensionalm¨obius-sturm theory.
Functional Analysis and Its Applications , 31(4):227–239, 1997.[2] B. Bertrand and E. Brugall´e. On the number of connected components of the paraboliccurve.
Comptes Rendus Math´ematique , 348(5):287–289, 2010.[3] E. Brugall´e and L. L´opez de Medrano. Inflection points of real and tropical plane curves.
Journal of Singularities , 2012.[4] A. Camacho-Calder´on.
Sobre la tropicalizaci´on de la propiedades las curvas Hessianas .PhD thesis, Universidad Nacional Aut´onoma de M´exico, 2018.[5] L. I. Hern´andez-Mart´ınez, A. Ortiz-Rodr´ıguez, and F. S´anchez-Bringas. On the affinegeometry of the graph of a real polynomial.
Journal of dynamical and control systems ,18(4):455–465, 2012.[6] L. I. Hern´andez-Mart´ınez, A. Ortiz-Rodr´ıguez, and F. S´anchez-Bringas. On the Hessiangeometry of a real polynomial hyperbolic near infinity.
Advances in Geometry , 13(2):277–292, 2013.[7] M. W. Hirsch.
Differential topology, volume 33 of Graduate Texts in Mathematics .Springer-Verlag, New York, 1994. 198] V. S. Kulikov. Calculation of singularities of an imbedding of a generic algebraic surfacein projective space P . Functional Analysis and Its Applications , 17(3):176–186, 1983.[9] E. E. Landis. Tangential singularities.
Funct. Anal. Appl. , 15:103–114., 1981.[10] L. L´opez de Medrano.
Courbe totale des hypersurfaces alg´ebriques r´eelles et patchwork .PhD thesis, 2006.[11] A. Ortiz-Rodr´ıguez.
G´eom´etrie diff´erentielle projective des surfaces alg´ebriques r´eelles .PhD thesis, Paris 7, 2002.[12] A. Ortiz-Rodr´ıguez. Quelques aspects sur la g´eom´etrie des surfaces alg´ebriques r´eelles.
Bulletin des Sciences math´ematiques , 127(2):149–177, 2003.[13] O. A. Platonova. Singularities of the mutual disposition of a surface and a line.
RussianMathematical Surveys , 36(1):248–249, 1981.[14] G. Salmon.
A treatise on the analytic geometry of three dimensions . Hodges, Smith, andCompany, 1865.[15] O. Y. Viro. Introduction to topology of real algebraic varieties.
Appears on the internetat http://archive.schools.cimpa.info/archivesecoles/20141218145605/es2007.pdf .[16] O. Y. Viro.
Topology: General and Algebraic Topology, and Applications Proceedings ofthe International Topological Conference held in Leningrad, August 23–27, 1982 , chapterGluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, pages187–200. Springer Berlin Heidelberg, Berlin, Heidelberg, 1984.[17] O. Y. Viro. Patchworking real algebraic varieties.
Uppsala University. Department ofMathematics , 1994.[18] Wolfram Research, Inc.