aa r X i v : . [ m a t h . AG ] D ec SURFACES WITH MANY SOLITARY POINTS
ERWAN BRUGALL´E AND OLIVER LABS
Abstract.
It is classically known that a real cubic surface in R P cannot havemore than one solitary point (or A • -singularity, locally given by x + y + z =0) whereas it can have up to four nodes (or A − -singularities, locally given by x + y − z = 0). We show that on any surface of degree d ≥ R P the maximum possible number of solitary points is strictly smaller than themaximum possible number of nodes.Conversely, we adapt a construction of Chmutov to obtain surfaces withmany solitary points by using a refined version of Brusotti’s Theorem. Com-bining lower and upper bounds, we deduce: d + o ( d ) ≤ µ ( A • , d ) ≤ d + o ( d ), where µ ( A • , d ) denotes the maximum possible number of soli-tary points on a real surface of degree d in R P . Finally, we adapt thisconstruction to get real algebraic surfaces in R P with many singular pointsof type A • k − for all k ≥ Introduction An ordinary double point , or A -singularity, of a hypersurface f = 0 in R P n or C P n is a non-degenerate singular point p of f ; i.e. f and all its partial derivativesvanish at p , but the hessian matrix H f ( p ) = ( ∂ f / ∂x i ∂x j ( p )) i,j =0 ...n is of rank n . In R P , there are exactly two real types of ordinary double points: we call theones which can be given locally by the affine equation x + y − z = 0 nodes or A − -singularities , and the others, locally given by x + y + z = 0, solitary ordinarydouble points , A • -singularities , or solitary points for short.[BLvS05] showed by construction that for large degree d the currently knownmaximum number of complex singularities on a surface of degree d in C P [Chm92]can also be achieved with a real surface with only real singularities. All real singu-larities appearing in their construction are nodes. In the present paper, we considersolitary points instead.We denote the maximum possible number of complex A -singularities on a com-plex hypersurface of degree d in C P by µ ( A , d ), and similarly for the real A − -and A • -singularities on real surfaces in R P : µ ( A − , d ) , µ ( A • , d ). Question 1.
It is clear that the maximum possible number of complex ordinarydouble points is at least as large as the corresponding real numbers: µ ( A − , d ) , µ ( A • , d ) ≤ µ ( A , d ) . Is any of these inequalities strict?
Date : November 27, 2018.2000
Mathematics Subject Classification.
Primary 14J17, 14J70; Secondary 14P25.
Key words and phrases. algebraic geometry, many real singularities, real algebraic surfaces.
Question 2.
Classical results on cubic surfaces [Sch63] and quartic surfaces [Roh13] in R P show that we have: µ ( A • ,
3) = 1 < µ ( A − , and µ ( A • ,
4) = 10 <
16 = µ ( A − , . These results suggest that it might be more difficult to have many solitary points onsurfaces than to have many nodes. Is this true for all d ≥ ? In this article, we answer those questions involving solitary points affirmativelyin Theorem 13: If d ≥ µ ( A • , d ) < µ ( A − , d ) , µ ( A , d ) . However the case of real nodes remains open in general although it is clear that µ ( A − , d ) ≤ µ ( A , d ) for all d . In fact, µ ( A − , d ) = µ ( A , d ) is only known for d = 1 , , . . . , µ ( A • , d ) is still far from the best knownupper bound. In the third section of this article, we improve the previously knownmaximum number of solitary points on a surface of degree d in R P by adaptinga construction of Chmutov and by using Brusotti’s Theorem. Altogether, we showfor d ∈ N by combining lower bound (Theorem 14) and upper bound (Corollary12): 14 d + o ( d ) ≤ µ ( A • , d ) ≤ d + o ( d ) . Together with the known cases in low degree, we get table 1 which provides anoverview of the known bounds for the maximum possible number of both variantsof the real ordinary double points. In that table, the upper bounds for the caseof A − -singularities are simply the complex ones most of which are due to Miyaoka[Miy84], the asymptotic lower bound was found in [BLvS05].degree d dµ ( A • , d ) ≥ ≈ d µ ( A • , d ) ≤ ≈ d µ ( A − , d ) ≥ ≈ d µ ( A − , d ) ≤ ≈ d Table 1.
An overview of the known bounds for the maximumpossible number of both variants of the real ordinary double pointson surfaces of degree d in R P : solitary points and nodes.More generally, an A j -singularity of a complex surface in C P is a singular pointlocally given by the equation x j +1 + y + z = 0. If k ≥ k = 1) real types of A k − -singularities, and we call the one given locally bythe real equation x k + y + z = 0 an A • k − -point . In section 4, we explain howto adapt our method to construct real surfaces in R P with many A • k − -points.More precisely, we prove that (see Proposition 20) for k, d ≥ k − d + o ( d ) ≤ µ ( A • k − , d ) ≤ k k − d + o ( d ) . URFACES WITH MANY SOLITARY POINTS 3
The upper bound is again Miyaoka’s bound on the number of complex A k − -pointsof a complex surface of degree d in C P . Acknowledgements:
We are grateful to Fr´ed´eric Bihan, Michel Coste, IliaItenberg, and Jean Jacque Risler for valuable and stimulating discussions.1.
Plane Curves with Solitary Points
In our results on real surfaces with solitary points we will use some facts aboutreal plane curves with solitary points. So, we give a brief overview about thisclassical subject. As in the case of A -singularities of surfaces mentioned in theintroduction, there are exactly two real types of ordinary double points on a realplane curve, also denoted by A • resp. A − .1.1. Nodes.
The value µ ( A − , d ) of the maximum possible number of nodes on areal plane curve of degree d has been known for a long time: µ ( A − , d ) = d ( d − . The upper bound is a consequence of the genus formula, and a generic configurationof d lines shows that this upper bound is sharp. The genus formula also shows thatthis bound can only be achieved with arrangements of d real lines no three of whichmeet in a point.There is a classical theorem, the Brusotti Theorem, which shows that we cansmooth each of the ordinary double points of a plane curve independently. Appliedto the d generic lines in the plane mentioned above, we may deduce that for anyinteger r between 0 and d ( d − , there is a real plane curve of degree d in R P withexactly r nodes as its only singularities.Let us denote by C ( d ) (resp. R C ( d )) the space of complex (resp. real) algebraiccurves of degree d in C P (resp. R P ). These are projective spaces of dimension d ( d +3)2 . Brusotti’s result is the following: Theorem 1 (Brusotti Theorem, usual formulation) . Let C be a real algebraic curveof degree d in R P with ordinary double points as its only singularities. For anyof these singularities, choose a local deformation. Then it is possible to vary thecurve C in the space R C ( d ) in such a way that all previously chosen deformationsare realized. This is the form of the theorem which is usually given because it can be appliedvery easily. It is a straightforward corollary of the following result which will bemore convenient for our purposes.
Theorem 2 (Brusotti Theorem, for a proof see e.g. [BR90]) . Let C be a complexalgebraic curve of degree d in C P with ordinary double points p , . . . , p k as its onlysingularities. Then there exists a small neighborhood V i of p i in C P for each i ,and a small neighborhood V of C in C ( d ) such that the analytic sets S i = { e C ∈ V | e C is non-singular except at some point in V i where it has an A } are all non-singular and intersect transversely. Moreover, the tangent space of S i at C is { e C ∈ C ( d ) | p i ∈ e C } . ERWAN BRUGALL´E AND OLIVER LABS
Solitary Points.
For solitary points, things are a bit more complicated, andthe exact maximum number of solitary points is only known since the 80’s.
Proposition 3.
Let d ∈ N . Then: µ ( A • , d ) ≤ ( d − d − . Proof.
According to Harnack’s Theorem, a non-singular real algebraic curve ofdegree d in R P has at most ( d − d − + 1 connected components. Now the resultfollows from the Brusotti Theorem. (cid:3) In most cases, this upper bound can be refined using the Petrovskii inequality(see [Pet33] or [Vir84]):
Proposition 4. If , = d ∈ N then: µ ( A • , d ) ≤ ( d − d − . Proof.
This bound is trivial for curves of odd degree, as one component of the curveis not contractible in R P . The Petrovskii inequality for plane curves implies thatif a curve of degree d = 2 k has ( d − d − + 1 ovals, then at least one of themcontains another oval if k ≥ (cid:3) The union of two complex conjugated lines is a real conic with one A • -point. If P ( x, y ) = 0 and P ( x, y ) = 0 are real equations of two real conics intersecting infour real points, then the real quartic with equation P ( x, y ) + P ( x, y ) = 0 hasfour A • -points. Hence, one has µ ( A • ,
2) = 1 and µ ( A • ,
4) = 4 (i.e. Proposition 3is sharp in degree 2 and 4). The proof that the upper bound given in Proposition4 is sharp for any other degree has first been given by Viro in the 80’s. As Viro’soriginal proof was not available to us, we sketch Kenyon and Okounkov’s [KO06]here:
Theorem 5 (Viro, see [Vir83], [KO06], or see [Shu93] for a proof of a more generalcase) . If d = 2 , then: µ ( A • , d ) = ( d − d − . Proof.
Let ε be a primitive d th root of unity, and define the polynomial e P d ( x, y ) = Q di,j =1 ( ε i x + ε j y + 1). Then, e P d ( x, y ) = P d ( x d , y d ) where P d ( x, y ) is a real polyno-mial of degree d , and the curve with equation P d ( x, y ) = 0 has exactly ( d − d − real solitary non-degenerate double points. (cid:3) As in the case of nodes, it follows from Brusotti’s Theorem that for any integer r between 0 and ( d − d − , there exists a real algebraic plane curve of degree d in R P with exactly r solitary nodes as its only singularities.For what follows, we need to introduce a distinction among solitary points of areal algebraic curve P ( x, y ) = 0 of even degree in C : those who are local minimaof the function ( x, y ) P ( x, y ) and those who are local maxima. Proposition 6 ([KO06],[Mik00]) . Let d ≥ be even and let P d ( x, y ) = 0 be thepolynomial constructed in the proof of Proposition 5. Then d ( d − solitary points ofthe curve P ( x, y ) = 0 are local maxima of the polynomial P ( x, y ) and the ( d − d − other solitary points of the curve P ( x, y ) = 0 are local minima of the polynomial P ( x, y ) .Proof. Kenyon’s and Okounkov’s polynomial P d ( x, y ) defines a Harnack curve, andMikhalkin showed that these curves have the desired property. (cid:3) Surfaces in R P : Upper Bounds In order to prove the upper bounds mentioned in the introduction, we need —in analogy to the Brusotti Theorem in the case of plane curves — a result aboutsmoothings of algebraic varieties.By a smoothing of a singular (real) algebraic hypersurface X of degree d in C P n , we mean a small perturbation of the coefficients of X such that the result is anon-singular (real) algebraic hypersurface of degree d in C P n . The Coste-HironakaTheorem now says that one can always smooth a real projective hypersurface insuch a way that no connected component disappears into the complex world: Theorem 7 (Coste-Hironaka, [Cos92]) . Let X be a singular real algebraic hyper-surface in R P n . Then there is a smoothing e X of X such that b ( X ) ≤ b ( e X ) , where b denotes the th Betti number, i.e. the number of connected components.
Remark 8.
In the special case of hypersurfaces with only solitary (ordinary double!)points as singularities, it is easy to prove this result using the construction givenin the paper [Cos92] : indeed, let P ( X , . . . , X n ) = 0 be the equation of such ahypersurface in R P n which does not have a singularity in the point (1 : 0 : · · · : 0) (which we may assume after a suitable change of coordinates). Then define e P ( X , . . . , X n ) := P ( X , . . . , X n ) + n X i =1 ε i X i ∂P∂X i ( X , . . . , X n ) . Each singular point p of P = 0 will still be a point on e P = 0 . Moreover, a shortcomputation shows that there are ε i small enough such that e P = 0 is non-singularin p because of the hessian criterion for A -singularities. But this means that ifthe ε i are small enough then near each solitary point p , the hypersurface P = 0 issmoothed into a small connected component of e P = 0 homeomorphic to an n -sphereand containing p . As we know the homology of projective non-singular complex algebraic hyper-surfaces, the Coste-Hironaka Theorem 7 combined with Smith Theory (see [Bre72])implies the following corollary:
Corollary 9 (Coste-Hironaka, [Cos92]) . Let X be a (possibly singular) real alge-braic hypersurface of degree d in R P n . Then b ( X ) ≤ (cid:18) ( d + 1) n +1 − ( − n +1 d + n − ( − n (cid:19) . In the case of projective surfaces in R P one can improve the upper bound onthe number of connected components thanks to the Petrovskii-Oleinik inequality(see, e.g., [DK00]): ERWAN BRUGALL´E AND OLIVER LABS
Corollary 10.
Let S be a (possibly singular) real algebraic surface of degree d in R P . Then b ( S ) ≤ d − d + 25 d . Remark 11.
Note that starting from degree 5, the maximal possible value of b ( S ) when S is a real algebraic surface of degree d is still unknown. Applying Corollary 10 in the special case of real surfaces with solitary doublepoints in R P , we get: Corollary 12.
For d ∈ N , we have: µ ( A • , d ) ≤ j d − d +25 d k , d even, j d − d +25 d k − , d odd.Proof. In odd degree, we can subtract one because in that case at least one of theconnected components from Corollary 10 is not homeomorphic to a sphere. (cid:3)
Comparing this upper bound with the lower bound obtained in the case of nodes(see Theorem 2 in [BLvS05] for a detailed formula) which is given by µ ( A − , d ) ≥ d − d + o ( d ) , d even, d − d + o ( d ) , d odd,we may deduce that one cannot reach the maximum number of nodes with surfaceshaving only solitary points: Theorem 13.
The maximum possible number of solitary points on a surface ofdegree d, d ≥ , in R P is strictly smaller than the maximum possible number ofnodes: µ ( A • , d ) < µ ( A − , d ) , µ ( A , d ) . We already mentioned in the introduction that this result is not very surprisingbecause it has been known for degree 3 and 4 for a long time. However, noticethat the corresponding statement in the case of plane curves does not hold: themaximum number of nodes on an irreducible curve of degree d equals the maximumnumber of solitary points on an irreducible curve of degree d : in both cases, it isthe genus of a smooth plane curve of degree d , as mentioned earlier.3. Surfaces in R P : Lower Bounds by Constructions In the preceding section we have shown that the maximum possible number ofsolitary points on a surface in R P is less than the corresponding number of nodes.Here we improve the currently known maximum number of solitary points. Indeed,in this section we show: Theorem 14.
Let d ∈ N . Then: µ ( A • , d ) ≥ ( d − d − d +4)8 if d is even, µ ( A • , d ) ≥ ( d − ( d − if d is odd . URFACES WITH MANY SOLITARY POINTS 7
We prove Theorem 14 in section 3.3. It is based on Chmutov’s method toconstruct singular complex surfaces. We thus explain this method first. Then wediscuss how to adapt it to obtain real algebraic surfaces with solitary nodes; finally,we show in section 3.4 that our result is asymptotically the best that one can achieveusing Chmutov’s method.3.1.
Known Constructions.
Notice that the previously best known lower boundfor the maximum possible number µ ( A • , d ) of solitary points on a surface in R P is far below d .Not many constructions are known. Certainly the most sophisticated one isShustin’s variant of Viro’s patchworking method for the singular case. This methodyields the optimal result in the case of plane curves, but already for complex surfacesin C P it only yields µ ( A , d ) ≥ d + o ( d ) (see [SW04]).There is another construction which is natural to consider: we take a polynomial P d ( x, y ) of degree d in two variables and we set f ( x, y, z ) = P d ( x, y ) + g ( z ) , where g ( z ) is a polynomial function of degree d in one variable z with the maximumpossible number ⌈ d ⌉ of local maxima z i with value g ( z i ) = 0. An even solitary point of an affine plane curve given by the equation P ( x, y ) = 0 is a solitary point ( x, y )of the curve P ( x, y ) = 0 which is a local minimum of the polynomial P ( x, y ), i.e.locally at p the graph of P ( x, y ) looks like z = x + y . We denote by es( P d ) thenumber of even solitary points of the curve P d ( x, y ) = 0. With these preliminaries,it is clear that the surface f ( x, y, z ) = 0 has ⌈ d/ ⌉ · es( P d ) solitary points: for eacheven solitary point ( a, b ) of the affine plane curve P d ( x, y ) = 0, we thus get a point( a, b, z i ) of f ( x, y, z ) = 0 which is locally of the form x + y + z . However, it iswell known that for a curve of degree d , one has (see [Vir84])es( P d ) ≤ d + o ( d ) , so one cannot expect to construct in this way surfaces of degree d with more than d + o ( d ) solitary points. Combining this method with Chmutov’s method weimprove the leading coefficient .3.2. Chmutov’s method.
We describe briefly how Chmutov constructed surfaceswith many (complex) ordinary double points in the 90’s [Chm92]. It is similar tothe idea mentioned in the previous paragraph. Despite its simplicity, Chmutov’ssurfaces still yield the best known lower bound for the maximum number of ordinarydouble points on a complex surface of degree d ≥
13. The best known lower boundin the case of real nodes ( A − -singularities) which we mentioned above and whichequals the current lower bound in the complex case is an adaption of Chmutov’sconstruction to real nodes [BLvS05]. So, it is quite natural to try to adapt themethod to solitary points. However, we will see that this process is not completelystraightforward, and we will need a refined version of Brusotti’s Theorem to makeit work.3.2.1. Chmutov’s Constructions.
Let T d ( z ) ∈ R [ z ] be the Tchebychev polynomial ofdegree d with ⌈ d − ⌉ extremal points with value − ⌊ d − ⌋ with value +1. Thiscan either be defined recursively by T ( z ) := 1, T ( z ) := z , T d ( z ) := 2 · z · T d − ( z ) − T d − ( z ) for d ≥
2, or implicitly by T d (cos( z )) = cos( dz ). In [Chm92], Chmutov usedthe Tchebychev polynomials to construct surfaces in C P with ≈ d (complex) ERWAN BRUGALL´E AND OLIVER LABS nodes using the so-called folding polynomials F A d ( x, y ) ∈ R [ x, y ] associated to theroot-system A : Chm A d ( x, y, z ) := F A d ( x, y ) + 12 ( T d ( z ) + 1) . The polynomials F A d ( x, y ) have critical points with only three different criticalvalues: 0, −
1, and 8. The surface Chm A d ( x, y, z ) = 0 is singular exactly at thosepoints at which the critical values of F A d ( x, y ) and ( T d ( z ) + 1) sum up to zero(i.e., either both are 0 or the first one is − Adaption to Real Nodes.
In [BLvS05], this construction was modified to yieldreal surfaces Chm A R ,d ( x, y, z ) = 0 with real nodes as singularities by using the so-called real folding polynomials F A R ,d ( x, y ) := F A d ( x + iy, x − iy ) , where i is the imaginary number. It is not difficult to see that the singularities ofthe surface Chm A R ,d ( x, y, z ) = 0 are indeed nodes, i.e. of type A − , by using the factthat the plane curve F A R ,d ( x, y ) = 0 is actually a product of d real lines no three ofwhich meet in a common point.3.2.3. Adaption to Solitary Points.
From the explanations in the previous para-graphs it is clear how to adapt Chmutov’s construction to yield solitary points: weneed to show the existence of a real polynomial f ( x, y ) with many local minimawith value +1 and local maxima with value −
1. More precisely:
Proposition 15.
Let f ( x, y ) be a real polynomial of even (resp. odd) degree d with α local minima with value +1 , and β local maxima with value − . Then the affinesurface defined by f ( x, y ) − T d ( z ) = 0 has ( α · d + β · ( d − (resp. ( α + β ) · ( d − )solitary points. The corresponding projective surface in R P has at most O ( d ) additional singularities. Notice that we cannot use a product of real lines such as F A R ,d ( x, y ) as the poly-nomial f ( x, y ) in order to obtain many solitary points because it has the wrongcritical values: the minima have critical value − Proof of the Lower Bound of Theorem 14.
We are now ready to proveTheorem 14 on the lower bound for µ ( A • , d ). According to Chmutov’s constructionand in particular Proposition 15, we have to construct polynomials in 2 variableswhose graphs have many minima with value +1 and many maxima with value − (cid:3) Proposition 16.
Let d ∈ N and P ( x, y ) be a real polynomial of degree d with α (resp. β ) local minima (resp. local maxima) with value 0. Then, there exists a realpolynomial in two variables of degree d with α local minima with value +1 and β local maxima with value − .Proof. We start with the following observation: if f ( x, y ) is a polynomial of degree d , then the graph of f , defined by the equation f ( x, y ) − z = 0, is a special linein the space C ( d ) of plane curves of degree d . Indeed, if f ( x, y ) = P a i,j x i y j , thenthe section of the graph of f by the hyperplane z = t is given by the equation URFACES WITH MANY SOLITARY POINTS 9 P a i,j x i y j − t = 0. If ( a , : a , : a , : · · · : a ,d ) are the coordinates in the space C ( d ), then the graph of f can be parameterized by the line t ( a , − t : a , : a , : · · · : a ,d ). For t = ∞ , this line passes through the point (1 : 0 : 0 : · · · : 0) whichrepresents the multiple line z d . Conversely, any line in the space of plane curves ofdegree d passing through the point (1 : 0 : 0 : · · · : 0) admits a parameterizationof the form t ( a , − t : a , : a , : · · · : a ,d ) which defines a polynomial f ( x, y ) = P a i,j x i y j of degree d .Let us go back to the polynomial P ( x, y ) of the Theorem. By assumption, thecurve defined by P has α + β solitary points. Now we show that we can perturbthe polynomial P ( x, y ) in such a way that all local minima (resp. maxima) stay onthe same level a (resp. b ) with a > b (see Figure 1). z=az=bz=az=bz=0 or Figure 1.
Two ways to perturb P ( x, y ).For any solitary point p of the curve P ( x, y ) = 0, we choose a small neighborhood V ( p ) of p in R P such that V ( p ) ∩ V ( q ) = ∅ if q = p is another solitary point. Wedenote by M ( P ) (resp. m ( P )) the set of solitary points of the curve P ( x, y ) = 0corresponding to local maxima (resp. minima) of P ( x, y ). Moreover, we denote byΣ M ( P ) (resp. Σ m ( P ) ) the stratum of real algebraic plane curves in C ( d ) in a smallneighborhood of P ( x, y ) = 0 with one solitary point in V ( p ) for any p ∈ M ( P ) (resp. m ( P )). Then, according to Brusotti’s Theorem, Σ M ( P ) and Σ m ( P ) are smooth andintersect transversely at the curve P ( x, y ) = 0. Moreover, we have:codim(Σ M ( P ) ) = β, codim(Σ m ( P ) ) = α and codim(Σ M ( P ) ∩ Σ m ( P ) ) = β + α ≤ ( d − d − d ( d + 3)2 − (3 d − . One can suppose that α > β > L the line in the space C ( d ) passing through the curve P ( x, y ) = 0 and z d . By a simple dimension computation, we prove that we can perturb L to a line e L still passing through z d = 0 and intersecting the stratum Σ M ( P ) and Σ m ( P ) oneafter the other:Define the projection π : R P d ( d +3)2 −→ R P d ( d +3)2 − ( a , : a , : a , : · · · ) ( a , : a , : · · · ) . None of the tangent spaces of Σ M ( P ) and Σ m ( P ) contains the point z d = 0, so π (cid:0) Σ M ( P ) (cid:1) and π (cid:0) Σ m ( P ) (cid:1) are non-singular and intersect transversely. Hence, wehave: codim( π (cid:0) Σ M ( P ) (cid:1) ) = β − , codim( π (cid:0) Σ m ( P ) (cid:1) ) = α − , codim( π (cid:0) Σ M ( P ) ∩ Σ m ( P ) (cid:1) ) = β + α − π (cid:0) Σ M ( P ) (cid:1) ∩ π (cid:0) Σ m ( P ) (cid:1) ) = β + α − . So, we have one degree of freedom to move from π ( P ( x, y ) = 0) out of π (cid:0) Σ M ( P ) ∩ Σ m ( P ) (cid:1) staying in π (cid:0) Σ M ( P ) (cid:1) ∩ π (cid:0) Σ m ( P ) (cid:1) which means exactly that we can perturb L to a line e L still passing through z d = 0 and intersecting the stratum Σ M ( P ) andΣ m ( P ) one after the other.As π (cid:0) Σ M ( P ) (cid:1) ∩ π (cid:0) Σ m ( P ) (cid:1) \ π (cid:0) Σ M ( P ) ∩ Σ m ( P ) (cid:1) has two connected components,we have two possible choices to perturb L . One will correspond to move up (resp.down) the local maxima (resp. minima) and the other will correspond to move up(resp. down) the local minima (resp. maxima), see Figure 1. Choosing the latterpossibility, we prove the proposition. (cid:3) The proposition can be interpreted as a refined version of Brusotti’s Theoremin a special case. Indeed, it does not only show that we can perturb each solitarypoint of a real plane curve P ( x, y ) = 0 into one of the two topological possibilities,but it proves that we can in addition put all solitary points which are deformed inthe same topological way on the same level of P ( x, y ), i.e. transform the points intoextremal points of the graph of P ( x, y ) with the same value.3.4. Optimality of our Construction.
We now show that using Chmutov’smethod it is asymptotically not possible to improve our lower bound obtainedin Theorem 14. Let us denote by µ Ch ( d ) the maximal possible number of solitarypoints of a real algebraic surface of degree d in R P constructed using Chmutov’smethod. Proposition 17.
Let d ∈ N . Then: µ Ch ( d ) = 14 d + o ( d ) . Proof.
The result is an immediate corollary of Theorem 14, Proposition 15 and ofthe following Proposition 18. (cid:3)
Let us denote by µ extr ( d ) the maximum possible number of local extrema of areal polynomial f ( x, y ) of degree d . We believe that the bound we establish now isknown, but as we did not find a reference for it, we include a proof here: URFACES WITH MANY SOLITARY POINTS 11
Proposition 18.
With the notation of the preceding proof, we have: µ extr ( d ) ≤ d + o ( d ) . Proof.
Denote by h the height function ( x, y, z ) z defined on R . Let f ( x, y ) bea real polynomial of degree d and denote by ν ( f ) (resp. ν ( f )) the number of localextrema (resp. hyperbolic critical points) of f . Consider a very large ball B in R containing all critical points of f , and consider D ( f ) the intersection of the graphof f with the cylinder with base B . Then, one can glue in R a disk to D ( f ) alongits border ∂D ( f ) adding a number of critical points for h which is at most linearin d . Then, we obtain a sphere S and h defines a Morse function on it. Hence, wehave ν ( f ) − ν ( f ) ≤ ad with a some integer number.On the other hand, the number of real critical points of f is not more thanits number of complex critical points, which is equal to ( d − . Taking all thistogether, we get: µ extr ( d ) ≤ d + o ( d ) . (cid:3) Higher singularities
Proposition 15 can also be applied to construct real algebraic surfaces in R P with many A • k − singularities. The method is exactly the same as in section 3.2.3,but instead of Tchebychev polynomials, we use polynomials with very degeneratecritical points of critical values ±
1. The existence of such polynomials is guaran-teed by applying the real version of Dessins d’Enfants (e.g. see [Bru06]) to theconstruction in [Lab06b].
Lemma 19.
Let d, k ≥ . Then there is a real polynomial T kd ( z ) of degree d with (cid:2) d − k − (cid:3) local maxima (resp. minima) which are critical points of multiplicity k − (resp. non-degenerate critical points) and with value +1 (resp. − ). Proposition 20.
Let k, d ≥ . We have: k − d + o ( d ) ≤ µ ( A • k − , d ) ≤ k k − d + o ( d ) . Proof.
The upper bound is Miyaoka’s bound. Let f d ( x, y ) be a real polynomial ofdegree d with α local minima with value 1, with β local maxima with value − α + β = ( d − d − . According to Theorem 5 and Proposition 16,such a polynomial exists. The lower bound in the theorem is given by consideringthe surface with equation f d ( x, y ) − T kd ( z ) = 0. (cid:3) Remark 21.
Using the method “ P d ( x, y ) + g ( z ) =0” described in section 3.1, onecould expect to obtain better lower bounds for µ ( A • k − , d ) as soon as k ≥ . In thiscase, P d ( x, y ) = 0 should be a plane curve with many even A • k − -points. However,up to our knowledge, the currently known constructions give only (1) µ ( A • k − , d ) ≥ k d which provide lower bounds a bit worse than ours for µ ( A • k − , d ) . The lower bound(1) can be obtained by considering the polynomials T d ( x ) − e T kd ( y ) where T d ( x ) isthe Tchebychev polynomial of degree d and e T kd ( y ) is a polynomial of degree d whichhas ⌊ d k ⌋ local minima which are critical points of multiplicity k − and with value +1 . The existence of the polynomials e T kd ( y ) can be proved with the same techniqueas in Lemma 19. In [Wes03, Proposition 3.5] , Westenberger claims that µ ( A • k − , d ) ≥ k − d . However, his proof of this proposition uses [Wes03, Lemma 3.1] which is wrongfor solitary points. Indeed, this lemma states that there exists a real algebraic curvewith Newton polygon the quadrangle with vertices (0 , , (0 , , (1 , and (2 k − , ,and with one A • k − point. However, such a curve cannot exist due to the followingproposition. The case of A • -singularities is easy to verify by hand; for the generalstatement, we need some more work: Proposition 22.
For any k ≥ , no real algebraic curve with Newton polygon thequadrangle with vertices (0 , , (0 , , (1 , and (2 k − , can have an A • k − point.Proof. For brevity, we will use the notations, definitions and basic results of [Bru06,section 4]. Suppose that there exists a curve C contradicting the proposition.Without loss of generality, we can assume that the equation of C is y + P ( x ) y + x =0, where P ( x ) is a real univariate polynomial in x of degree 2 k −
1. The discriminantof C seen as a polynomial in y is R ( x ) = P ( x ) − x , and it is clear that the topologyof C can be recovered out of the root scheme realized by the polynomials P ( x ), Q ( x ) = − x , and R ( x ). One sees that R ( x ) > x ≤
0, and since C has an A • k − point, R ( x ) must have a root of order 2 k close to which R ( x ) is non positive.It follows that the polynomials P ( x ), Q ( x ) and R ( x ) realize the root scheme (cid:0) ( p, b ) , ( q, , ( r, a ) , ( p, b ) , ( r, a ) , ( p, b ) , . . . , ( r, a i ) , ( p, b i ) , ( r, k ) , ( p, b i +1 ) , ( r, a i +1 ) , ( p, b k ) , ( r, a k ) (cid:1) where i , k , a j , b and b j are some non negative integers, and a >
0. It is not hardto see from the real rational graphs (or Dessins d’Enfants) that this is equivalentto the existence of three real polynomials e P ( x ), e Q ( x ) and e R ( x ) of degree 4 k − (cid:0) ( r, , ( p, b ) , ( r, a − , ( p, b ) , ( r, a ) , ( p, b ) , . . . , ( r, a i ) , ( p, b i ) , ( r, k ) , ( p, b i +1 ) , ( r, a i +1 ) , ( p, b k ) , ( r, a k ) (cid:1) . But then, e Q ( x ) = − β with β a nonzero real number, and e R ( x ) = e P ( x ) − β =( e P ( x ) − β )( e P ( x ) + β ). Now, the polynomials e P ( x ) − β and e P ( x ) + β are relativelyprime and of degree 2 k −
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