We determine the expected curvature polynomial of random real projective
varieties given as the zero set of independent random polynomials with Gaussian
distribution, whose distribution is invariant under the action of the
orthogonal group. In particular, the expected Euler characteristic of such
random real projective varieties is found. This considerably extends previously
known results on the number of roots, the volume, and the Euler characteristic
of the solution set of random polynomial equationsComment: 38 pages. Version 2: corrected typos, changed some notation, rewrote
  proof of Theorem 5.

Buergisser, Peter

English

arXiv

Peter Bürgisser

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 345 (2007) 507–512http://france.elsevier.com/direct/CRASS1/Probability TheoryAverage Euler characteristic of random real algebraic varietiesPeter Bürgisser 1Institute of Mathematics, University of Paderborn, 33095 Paderborn, GermanyReceived 5 September 2007; accepted 24 September 2007Presented by Philippe G. CiarletAbstractWe determine the expected curvature polynomial of real projective varieties given as the zero set of independent random poly-nomials with Gaussian distribution, which is invariant under the orthogonal group. In particular, the expected Euler characteristicof such random real projective varieties is found. This considerably extends previously known results on the number of roots, thevolume, and the Euler characteristic of the real solution set of random polynomial equations. To cite this article: P. Bürgisser, C. R.Acad. Sci. Paris, Ser. I 345 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméLa caractéristique d’Euler moyenne des variétés algébriques réelles aléatoires. Dans cet article, nous déterminons l’espé-rance du polynôme de courbure d’une variété projective réelle qui est donnée comme ensemble de zéros de polynômes aléatoires,avec une distribution gaussienne qui est invariante par le groupe orthogonal. En particulier, nous explicitons la caractéristiqued’Euler de telles variétés projectives réelles aléatoires. Ces résultats généralisent considérablement la connaissance du nombre dezéros, du volume, et de la caractéristique d’Euler, des ensembles de zéro des systèmes de polynômes aléatoires. Pour citer cetarticle : P. Bürgisser, C. R. Acad. Sci. Paris, Ser. I 345 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeSoit Hd,n l’espace vectoriel des polynômes réels homogènes de degré d en les variables X0, . . . ,Xn. Soitf = ∑α fαXα00 · · ·Xαnn ∈ Hd,n avec des coefficients fα qui sont des variables aléatoires indépendantes gaussiennescentrées avec variance(dα) := d!α0!···αn! . On peut montrer que la distribution de probabilité induite sur Hd,n est inva-riante par le groupe orthogonal O(n+1). Nous dirons que f est distribuée selon Kostlan [7]. Considerons un systèmef1(x) = 0, . . . , fn(x) = 0 de tels polynômes aléatoires. Shub et Smale [14] ont montré que l’espérance du nombrede zéros dans l’espace projectif réel Pn est égale à√d1 · · ·dn, c’est-à-dire la racine carrée du produit des degrés despolynômes. Dans le cas où tous les polynômes ont le même degré, ce résultat a été également établi par Kostlan [7].E-mail address: pbuerg@upb.de.1 Partially supported by DFG grant BU 1371/2-1 and Paderborn Institute for Scientific Computation.1631-073X/$ – see front matter © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2007.10.013508 P. Bürgisser / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 507–512Dans le cas sous-determiné f1(x) = 0, . . . , fs(x) = 0 (s < n), l’ensemble des zéros Z(f1, . . . , fs) est une sous-variété de l’espace projectif réel Pn. Comme généralisation de la cardinalité on peut prendre le volume en dimensionn − s ou bien la caractéristique d’Euler. On a E vol(Z(f1, . . . , fs)) = √d1 · · ·ds vol(Pn−s) pour un système de po-lyômes fi distribué selon Kostlan [7].Podkorytov [11] a défini le paramètre δ(f ) := E‖Df (x)‖2nEf (x)2d’un polynôme aléatoire avec une distribution gaus-sienne centrée, qui est invariante par le groupe orthogonal O(n + 1). Ici x est un point quelconque dans Sn etDf (x) :TxSn → R est la dérivée de f interprété comme fonction f :Sn → R (noter que δ(f ) est indépendant de x parinvariance et homogénéité). Le paramètre d’un polynôme f ∈ Hd,n distribué selon Kostlan est égale au degré d . Enutilisant la théorie de Morse, Podkorytov [11] a montré que l’espérance de la caractéristique d’Euler de l’hypersurfaceZ(f ) ⊆ Pn peut être écrite commeEχ(Z(f )) = In(√δ(f ) )/In(1) si n est impair,où In(√δ) := ∫√δ0 (1 − x2)n−12 dx. Dans le cas où n est pair, nous remarquons que, presque sûrement, Z(f ) est unevariété différentielle compacte de dimension impaire ; alors sa caractéristique d’Euler est zéro.Nous étendons le résultat de Podkorytov au cas d’une codimension plus grande :Théorème 1.1. Soient f1 ∈ Hd1,n, . . . , fs ∈ Hds,n des polyômes indépendants avec une distribution gaussienne centréequi est invariante par le groupe orthogonal. Soit δi le paramètre de fi et supposons que s  n et que n − s est pair.Alors Eχ(Z(f1, . . . , fs)) = a0 + a2 + · · · + an−s2, où les aj sont les coefficients de la série de puissances suivante :∞∑j=0a2j T2j :=s∏i=1δ1/2i(1 − (1 − δi)T 2)1/2 .Dans le cas où tous les paramètres sont égaux à δ, nous avonsEχ(Z(f1, . . . , fs)) = δs/2n−s2∑k=0C(s)k (1 − δ)k = (−1)n−s2 C(s)n−s2δn2 + O(δ n2 −1) (δ → ∞)où C(s)0 = 1 et C(s)k = s(s+2)(s+4)···(s+2k−2)k!2k si k > 0.La preuve repose sur l’étude des coefficients de courbure de la variété différentielle formée par les solutions etsur la formule principale de la géometrie intégrale [5,4,6]. En fait, nous déterminons l’espérance du polynôme decourbure (Éq. (5)) qui contient la caractéristique d’Euler.1. IntroductionLet Hd,n denote the vector space of homogeneous real polynomials of degree d in the variables X0, . . . ,Xn. Letf = ∑α fαXα00 · · ·Xαnn ∈ Hd,n be such that the coefficients fα are independent centered Gaussian random variableswith variance(dα) := d!α0!···αn! . The induced probability distribution on Hd,n can be shown to be invariant under thenatural action of the orthogonal group O(n + 1). We will say that such f is a Kostlan distributed random polyno-mial [7]. Consider a system f1(x) = 0, . . . , fn(x) = 0 of such random polynomials. Shub and Smale [14] showed thatits expected number of real roots in real projective space Pn equals√d1 · · ·dn, i.e., the square root of the product ofthe degrees di of the polynomials. This result was also found by Kostlan [7] in the case where all polynomials havethe same degree. Extensions of this result can be found in [12,9,8]. A different proof of the Kostlan–Shub–Smaletheorem based on the Rice formula from the theory of random fields was recently given by Azaïs and Wschebor [2].Wschebor [17], for the first time, analyzed the variance of the number of real roots.Considerably less is known for the underdetermined case f1(x) = 0, . . . , fs(x) = 0 (s < n) where the zero setZ(f1, . . . , fs) is an algebraic subvariety of Pn. As a measure of its size different choices come to mind: one possiblechoice is the (n− s)-dimensional volume. Another generalization of cardinality to higher dimensional solutions sets isthe Euler characteristic. Both of these measures have been considered already. For a system of Kostlan distributed fi ,we have E vol(Z(f1, . . . , fs)) = √d1 · · ·ds vol(Pn−s), as was first observed in [7] for the case d1 = · · · = ds .P. Bürgisser / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 507–512 509Podkorytov [11] more generally investigated invariant centered Gaussian random polynomials f ∈ Hd,n (invariantmeaning that the distribution of f is invariant under the action of the orthogonal group O(n + 1)). He defined itsparameterδ(f ) := E‖Df (x)‖2nEf (x)2,where x denotes any point in Sn and Df (x) :TxSn → R is the derivative of f interpreted as a function f :Sn → R (wenote that δ(f ) is independent of x by invariance and homogeneity). The parameter of a Kostlan distributed f ∈ Hd,nequals the degree d .Using Morse theory, Podkorytov proved in [11] that the expected Euler characteristic of the hypersurface Z(f )in Pn can be expressed by its parameter as follows:Eχ(Z(f )) = In(√δ(f ) )/In(1) for odd n, (1)where In(√δ) := ∫√δ0 (1 − x2)n−12 dx. Note that if n is even, Z(f ) is almost surely a compact odd-dimensional mani-fold and therefore its Euler characteristic vanishes.We extend Podkorytov’s result to higher codimension:Theorem 1.1. Suppose that f1 ∈ Hd1,n, . . . , fs ∈ Hds,n are independent invariant centered Gaussian random poly-nomials and let δi denote the parameter of fi . Suppose that s  n and n − s is even. Then Eχ(Z(f1, . . . , fs)) =a0 + a2 + · · · + an−s2, where the aj are the coefficients of the following power series∞∑j=0a2j T2j :=s∏i=1δ1/2i(1 − (1 − δi)T 2)1/2 .In the case where all parameters equal δ, we more explicitly haveEχ(Z(f1, . . . , fs)) = δs/2n−s2∑k=0C(s)k (1 − δ)k = (−1)n−s2 C(s)n−s2δn2 + O(δ n2 −1) (δ → ∞),where C(s)0 = 1 and C(s)k = s(s+2)(s+4)···(s+2k−2)k!2k for k > 0.For proving this result, a direct use of Morse theory as in [11] does not seem feasible. Instead, it turned out to beessential to study the notions of curvature coefficients of the solution manifold and to employ the kinematic formulaof integral geometry [5,4,6]. In fact, we determine the expectation of the so-called curvature polynomial (Eq. (5)),from which the Euler characteristic can be easily read off.A different proof of Theorem 1.1, based on Weyl’s tube formula and the Rice formula from the theory of randomfields, was given in [3]. We finally remark that our work has connections with recent investigations on the geometryof random fields, cf. Adler and Taylor [15].2. Background from differential and integral geometryIn a seminal work, Weyl [16] derived a formula for the volume of the tube T (M,α) := {y ∈ Pn | dist(y,M)  α} ofradius α around an m-dimensional compact smooth submanifold M of Pn (actually Weyl considered the sphere Sn).Let s := n − m denote the codimension of M . Weyl [16] proved that, for sufficiently small α > 0, vol(T (M,α)) canbe written as a linear combinationvol(T (M,α)) = ∑0em,e evenKs+e(M)Jn,s+e(α)/2of the functions Jn,k(α) :=∫ α0 (sinρ)k−1(cosρ)n−k dρ. The coefficients Ks+e(M) depend on the curvature of M in Pnand will be thus called its curvature coefficients. Let On−1 := 2πn/2/Γ (n/2) denote the dimensional volume of theunit sphere Sn−1. Following Nijenhuis [10], we define the curvature polynomial of M by settingμ(M;T ) :=∑μe(M)Te :=∑ Ks+e(M)Om−eOs+e−1T e.0em,e even 0em,e even510 P. Bürgisser / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 507–512For example, a linear subspace Pm of Pn satisfies μ(Pm;T ) = 1. The constant term of the curvature polynomialdescribes the volume of M , namely μ0(M) = vol(M)/vol(Pm).The following statement can be deduced from the generalized Gauss–Bonnet theorem [1]:Theorem 2.1. Let M be a compact smooth submanifold of Pn of even dimension. Then the Euler characteristic of Mcan be expressed as χ(M) = μ(M;1).Nijenhuis [10] gave an elegant formulation of the principal kinematic formula of integral geometry [5,4] in termsof reduced polynomial multiplication of the curvature polynomials. He formulated this for Euclidean space, but itholds in much higher generality, cf. [13,6].Theorem 2.2. Let M and N be compact smooth submanifolds of Pn of the dimensions m and p, respectively, suchthat m + p  n. Then we have∫μ(M ∩ gN;T )dg ≡ μ(M;T )μ(N;T ) mod T m+p−n+1,where the integration is with respect to the Haar measure on the orthogonal group O(n + 1) scaled such that thevolume of O(n + 1) equals 1. In particular, we have∫μ(M ∩ gPp;T )dg ≡ μ(M;T ) mod T m+p−n+1.3. Proof of Theorem 1.1Podkorytov’s result (1) takes a simpler form when expressed in terms of generating functions. We define χk(δ) :=Eχ(Z(f )), where f ∈ Hd,2k+1 is an invariant centered Gaussian random polynomial with parameter δ.We haveI2k+1(1) =1∫0(1 − x2)k dx = 2kk!1 · 3 · 5 · · · (2k + 1) .The binomial series yields(1 − Y)−3/2 =∞∑k=01I2k+1(1)Y k.By a manipulation of formal power series we obtain from (1)∞∑k=0χk(δ)T2k =∞∑k=0I2k+1(δ1/2)I2k+1(1)T 2k =∞∑k=01I2k+1(1)√δ∫0(1 − x2)k dxT 2k=√δ∫0∞∑k=0((1 − x2)T 2)kI2k+1(1)dx =√δ∫0dx(1 − T 2 + x2T 2)3/2 .Usingξ∫0dx(A + Bx2)3/2 =ξA√A + Bξ2we arrive atχ(δ;T ) :=∞∑k=0χk(δ)T2k = δ1/2(1 − T 2)(1 − (1 − δ)T 2)1/2 . (2)We connect now the expected Euler characteristic to expected curvature coefficients:P. Bürgisser / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 507–512 511Lemma 3.1. Suppose that f ∈ Hd,n has O(n + 1)-invariant distribution with parameter δ. Then we haveEμ0(Z(f )) = χ0(δ) and Eμ2k(Z(f )) = χk(δ) − χk−1(δ) for 1  k  (n − 1)/2. In particular, Eμ2k(Z(f )) de-pends only on k and δ and not on the dimension n of the ambient space.Proof. We denote by f ′ and f ′′ the restrictions of f to R2k+2 and R2k , respectively. Theorem 2.1 implies thatχk(δ) =k∑i=0Eμ2i(Z(f ′)), χk−1(δ) =k−1∑i=0Eμ2i(Z(f ′′)). (3)We use the following result that is easily deduced from Theorem 2.2 by taking N = Pn′ :Lemma 3.2. Suppose f ∈ Hd,n has O(n + 1)-invariant distribution and n′  n. Then the restriction f ′ ∈ Hd,n′of f to Rn′+1 has O(n′ + 1)-invariant distribution and has the same parameter. For 0  e < n′, e even, we haveEμe(Z(f ′)) = Eμe(Z(f )).This lemma implies Eμ2i (Z(f )) = Eμ2i (Z(f ′)) = Eμ2i (Z(f ′′)) for 0  i  k − 1 and Eμ2k(Z(f )) =Eμ2k(Z(f ′)). Subtracting the two equations (3), we obtain χk(δ)−χk−1(δ) = Eμ2k(Z(f )) as claimed. The assertionfor k = 0 is obvious. By Lemma 3.1 the formal power seriesμ(δ;T ) :=∞∑k=0Eμ2k(Z(f ))T 2ksatisfies μ(δ;T ) = (1 − T 2)χ(δ;T ). Eq. (2) implies thatEμ(δ;T ) = δ1/2(1 − (1 − δ)T 2)−1/2. (4)Suppose now that fi ∈ Hdi,n are as in Theorem 1.1. Write f := (f1, . . . , fs−1). Theorem 2.2 tells us that for almostall f ,fs∫μ(Z(f ) ∩ gZ(fs);T)dg ≡ μ(Z(f );T )μ(Z(fs);T )modT n−s+1,where the integral is over O(n+ 1) with respect to the Haar measure scaled to 1. We take the expectation with respectto f and fs . Taking their independence into account we get∫Ef ,fs μ(Z(f ) ∩ gZ(fs);T)dg ≡ Eμ(Z(f );T )Eμ(Z(fs);T )modT n−s+1.By invariance, the integrand does not depend on g and equals Eμ(Z(f1, . . . , fs);T ). We conclude by induction thatEμ(Z(f1, . . . , fs);T) ≡ Eμ(Z(f1);T ) · · ·Eμ(Z(fs);T ) mod T n−s+1.Plugging in the explicit expression (4) for Eμ(Z(fi);T ) we getEμ(Z(f1, . . . , fs);T) ≡s∏i=1δ1/2i(1 − (1 − δi)T 2)1/2 mod Tn−s+1. (5)The claim about the Euler characteristic follows now by Theorem 2.1. Eq. (5) is of independent interest and can be seen as the main result of this work. By evaluating this equation atT = 0 we getEμ0(Z(f1, . . . , fs)) = E vol(Z(f1, . . . , fs))/vol(Pn−s) = (δ1 · · · δs)1/2.We note that this is a generalization of the results by Kostlan [7] and Shub and Smale [14] since this holds for anyinvariant centered Gaussian random polynomials.512 P. Bürgisser / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 507–512AcknowledgementsI thank Martin Lotz and Mario Wschebor for useful discussions.References[1] C.B. Allendoerfer, A. Weil, The Gauss–Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. 53 (1943) 101–129.[2] J.-M. Azaïs, M. Wschebor, On the roots of a random system of equations. The theorem on Shub and Smale and some extensions, Found.Comput. Math. 5 (2) (2005) 125–144.[3] P. Bürgisser, Average volume, curvatures, and Euler characteristic of random real algebraic varieties, Preprint, Arxiv: math.PR/0606755,October 2006.[4] S.S. Chern, On the kinematic formula in integral geometry, J. Math. Mech. 16 (1966) 101–118.[5] H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418–491.[6] R. Howard, The kinematic formula in Riemannian homogeneous spaces, Mem. Amer. Math. Soc. 106 (509) (1993) vi+69.[7] E. Kostlan, On the distribution of roots of random polynomials, in: The Work of Smale in Differential Topology, From Topology to Compu-tation: Proceedings of the Smalefest, Springer, 1993, pp. 419–431.[8] G. Malajovich, J.M. Rojas, High probability analysis of the condition number of sparse polynomial systems, Theoret. Comput. Sci. 315 (2–3)(2004) 524–555.[9] A. McLennan, The expected number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (1) (2002)49–73.[10] A. Nijenhuis, On Chern’s kinematic formula in integral geometry, J. Differential Geometry 9 (1974) 475–482.[11] S.S. Podkorytov, The mean value of the Euler characteristic of an algebraic hypersurface, Algebra i Analiz 11 (5) (1999) 185–193. Englishtranslation: St. Petersburg Math. J. 11 (5) (2000) 853–860.[12] J.M. Rojas, On the average number of real roots of certain random sparse polynomial systems, in: The Mathematics of Numerical Analysis,Park City, UT, 1995, in: Lectures in Appl. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1996, pp. 689–699.[13] L.A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley Publishing Co., Reading, MA, 1976.[14] M. Shub, S. Smale, Complexity of Bézout’s theorem II: volumes and probabilities, in: F. Eyssette, A. Galligo (Eds.), Computational AlgebraicGeometry, in: Progress in Mathematics, vol. 109, Birkhäuser, 1993, pp. 267–285.[15] J.E. Taylor, R.J. Adler, Euler characteristics for Gaussian fields on manifolds, Ann. Probab. 31 (2) (2003) 533–563.[16] H. Weyl, On the volume of tubes, Amer. J. Math. 61 (2) (1939) 461–472.[17] M. Wschebor, On the Kostlan–Shub–Smale model for random polynomial systems. Variance of the number of roots, J. Complexity 21 (6)(2005) 773–789.

Comptes Rendus Mathematique

Average Euler characteristic of random real algebraic varieties

Bürgisser, Peter

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2007.10.013.pdf

Average volume, curvatures, and Euler characteristic of random real algebraic varieties

Abstract

Similar works

Full text

Available Versions

Comptes Rendus Mathématique

Numérisation de Documents Anciens Mathématiques