The topology of a random intersection of real quadrics

Transcription

1 The topology of a random intersection of real quadrics Antonio Lerario Cortona 2012 A. A. Agrachev s 60th birthday

2 Real random polynomials or equivalently f (t) = a 0 + a 1 t a d t d f (x 0, x 1 ) = a 0 x d 0 + a 1 x d 1 0 x a d x d 1 The coefficients of f are gaussian random variables, i.e.: P{a i c} = 1 c σ 2π e x 2 2σ i 2 dx, (a i N(0, σ i )) Question What is the expected value E d of the number of real roots of f?

3 The average number of real roots of a random polynomial a i N(0, 1), i = 0,... d, independent: lim d E d log d = 2 π (Kac, 43) a i N(0, ( d i ) ), i = 0,... d, independent: Theorem (Edelman and Kostlan, 95) E d = d What is the meaning of this distribution?

4 Gaussian distributions Gaussian distribution on a vector space V scalar product on V P{v A} = 1 e v,v 2 dv c V = H d,1 = {f homogeneous of degree d in two variables} f, g = f (z)g(z)e z 2 dz C 2 this gives the previous distribution, i.e. a i N(0, ( d i ) ). A

5 Weyl distribution H d,n = {f real, homogeneous of degree d in n + 1 variables} f, g = f (z)g(z)e z 2 dz C n+1 Definition (Weyl distribution) f (x) = α f α x α, α = (α 0,..., α n ) f α N(0, d! α 0! α d! )

6 Real zeroes vs Complex zeroes f H d,n : Z R (f ) RP n, Z C (f ) CP n Idea We wish to compare Z R (f ) and Z C (f ). f H d,1 : Z R (f ) Z C (f ) f Weyl distributed : E d = d d.

7 The expected volume More generally let X R = Z R (f 1,..., f k ) RP n and RP n with Fubiny-Study density induced from CP n : Vol(X R ) Vol(X C ) = d 1 d k Theorem If f 1,..., f k independent and Weyl distributed: EVol(X R ) = d 1,..., d k Vol(RP n k ) (Shub and Smale 00, Burgisser 07)

8 The curvature polynomial M RP n of dimension m T (M, ɛ) = {y RP n d(y, M) ɛ} For ɛ > 0 small enough (Weyl): Vol(T (M, ɛ)) = 0 e m, e even K s+e (M)J n,s+e (ɛ)/2 where J n,k (ɛ) = ɛ 0 (sin t)k 1 (cos t) n k dt and K s+e depend only on the intrinsic geometry of M (curvature coefficients). µ(m, x) = 0 e m, e even K s+e (M) Vol(S m e )Vol(S s+e 1 ) x e µ(m, 0) is the normalized volume and µ(m, 1) = χ(m) (m even). Burgisser has computed the expected curvature polynomial!

9 Number of points = zero dimensional volume EVol(X R ) = d 1,..., d k Vol(RP n k ) For k = n = 1 this is E d = d Idea Number of points = zero dimensional volume {real roots of f } {complex roots of f } generalizes as Vol(Z R (f )) Vol(Z C (f )).

10 Number of points = total Betti number b(x ) = b i (X ), b i (X ) = rkh i (X ) b i (X ) is the number of i + 1 dimensional holes in X. Idea Number of points in Z R (f ) = b(z R (f )). generalizes as {real roots of f } {complex roots of f } b(x R ) b(x C ) (Smith sinequality)

11 Smith s inequality b(x R ) b(x C ) for curves Example: X R real curve in RP 2 of degree d. Harnack: X R has at most (d 1)(d 2) 2 Genus formula: X C is a Riemann surface with g = b(x R ) = 2b 0 (X R ) 2g + 2 = b(x C ) + 1 ovals. (d 1)(d 2) 2 Curves with g + 1 ovals are called maximal and are extremely hard to built starting from their coefficients.

12 Average number of ovals of a real curve The answer is not known. Curves with approximatively d 2 components exponentially rarefact (Gayet and Welschinger, 11). E[b(X R )] lim π (Sarnak, 11) d d

13 Intersection of real quadrics From now on X R = Z R (q 1,..., q k ), q i H 2,n Idea To compare b(x R ) and b(x C ). b(x C ) is known (for regular intersections): k = 1 : b(x C ) = n (1 + ( 1)n+1 ) k = 2 : b(x C ) = 2n 1 2 (1 + ( 1)n+1 ) k = 3 : b(x C ) = n (5 + 3( 1)n ) = n 2 + O(n) What about X R?

14 Topology of intersection of real quadrics W = span{q 1,..., q k } Σ = {q ker(q) 0} { q 2 = 1} Σ W = Σ W Theorem (Agrachev 90, Agrachev and L. 11) b(x R ) b(σ W ) k = 1 : b(x R ) = 2 min i + W k = 2 : b(x R ) 2 min i + W + n b(σ W ) k = 3 : b(x R ) 2 min i + W + n b(σ W )

15 Complexity of intersection of real quadrics X R = Z R (q 1,..., q k ) Σ W S k 1 hypersurface of degree n + 1 (number of variables) k = 1 : Σ W = k = 2 : Σ W = points on S 1 k = 3 : Σ W = curve of degree n + 1 on S 2 k = 1 : b(σ W ) = 0 k = 2 : b(σ W ) 2n + 2 k = 3 : b(σ W ) n 2 + O(n) Theorem (Barvinok 99, Agrachev and L. 11) b(x R ) n O(k 1)

16 The topology of a random quadratic hypersurface in RP n q real quadratic form in n + 1 variables. q(x) = x, Qx, Q Sym n+1 (R) Idea q Weyl distributed is equivalent to Q GOE. b(z R (q)) = 2 min i + W W = span(q) hence min i + W = min{i + (q), i + ( q)} Question E min{i + (q), i + ( q)} =?

17 Wigner s semicircular law ESD : Theorem (Wigner) For every ψ C 0 c (R): µ n = 1 n n i=0 δ λi (Q)/ n SC : µ sc = 1 2π (4 x 2 ) 1/2 + dx lim E ψdµ n = n R R ψdµ sc Q GOE For every A R the expected number of eigenvalues of Q/ n in A divided by n is asymptotically as A dµ sc

18 The topology of a random quadratic hypersurface in RP n i + (q) = number of eigenvalues in [0, ) E min{i + (q), i + ( q)} lim = 1 n n 2 Theorem (L. 12) E[b(X R )] lim = 1 n n Since b(x C ) = n (1 + ( 1)n+1 ), then: [ ] lim E b(xr ) = 1 n b(x C ) Thus Smith s inequality b(x R ) b(x C ) is expected to be sharp as we let n growth.

19 The random intersection of two quadrics in RP n X R = Z R (q 1, q 2 ) b(x R ) = 2 min i + W + n b(σ W ) + ɛ, ɛ {0, 1, 2} Wigner s semicircular law: What about E min i + W lim n n = 1 2 Eb(Σ W ) lim =? n n

20 Gap probability in GOE Integral geometry formula gives: Eb(Σ W ) = Eckart-Young: d F (q, Σ) = σ(q) 1 Vol(Σ) Vol(S N 2 ) Vol(Σ) = lim ɛ 0 1 P{no eigenvalues in ( ɛ, ɛ)} 2ɛ Vol(Σ) = Γ( n 2 ) n(n+1) Γ( n 1 2 )Γ( 1 2 )Γ( ) 2 )Vol(S lim Eb(Σ Γ( n 2 W ) = lim ) Vol(S N 1 ) n n Γ( n 1 2 )Γ( 1 2 )Γ( 3 2 ) Vol(S N 2 ) = 0

21 The random intersection of two quadrics in RP n X R = Z R (q 1, q 2 ) b(x R ) = 2 min i + W + n b(σ W ) + ɛ, ɛ {0, 1, 2} lim n E min i + W n = 1 2 lim n Eb(Σ W ) n = 0 Theorem (L. 12) Let X R be the intersection of two real random quadrics in RP n, then: Eb(X R ) lim = 1 n 2n

22 Smith s inequality Since b(x C ) = 2n (1 + ( 1) n+1 ), then: [ ] lim E b(xr ) = 1 n b(x C ) Thus also in this case Smith s inequality b(x R ) b(x C ) is expected to be sharp as we let n growth. Open Question What happens with more quadrics?

23 Random Hilbert s 16th problem X R = Z R (q 1, q 2, q 3 ) b(x R ) b(σ W ) Σ W is the curve on S 2 given by det(x 1 Q 1 + x 2 Q 2 + x 3 Q 3 ) = 0 this curve has degree n + 1, hence: b(σ W ) n 2 + O(n) b(x C ) = n 2 + O(n) lim Eb(Σ W ) n n 2 = lim Eb(X R) n n 2 = lim Eb(X R) n b(x C ) = 1 Thus if Smith s inequality is expected to be sharp for X R, we can produce almost maximal curves with high probability!

24 Random Hilbert s 16th problem Consider the space W d of f : S 2 R spherical harmonic of degree d with the gaussian density given by: f, g = fgω S 2 S 2 Theorem (Nazarov and Sodin, 07 ) There exists c 1 > 0 such that for a random spherical harmonic of degree d: Eb(Z(f )) lim d d 2 = c 1

25 Bibliography A. A. Agrachev: Topology of quadratic maps and hessians of smooth maps, Itogi nauki. VINITI. Algebra. Topologiya. Geometriya, 1988, v.26, (Translated in J. Soviet Math. 49, 1990, no. 3, ). A. A. Agrachev, A. Lerario: Systems of quadratic inequalities, Prooceedings of the London Mathematical Society (to appear), arxiv: v2. A. I. Barvinok: On the Betti numbers of semialgebraic sets defined by few quadratic inequalities, Discrete and Computational Geometry, 22:1-18 (1999). P. Burgisser, Average Euler characteristic of random algebraic varieties, C. R. Acad. Sci. Paris, (2007). E. Edelman, A. Kostlan, How many zeros of a random polynomial are real?, Bulletin of the American Mathematical Society 32 (1995), D. Gayet, J-Y. Welschinger, Exponential rarefaction of real curves with many components, Publ. Math. Inst. Hautes tudes Sci. No. 113 (2011), M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. Volume 49, Number 4 (1943), A. Lerario, The total Betti number of the intersection of two real random quadrics, arxiv: v1. F. Nazarov, M. Sodin, On the Number of Nodal Domains of Random Spherical Harmonics, arxiv: v1 P. Sarnak, Letter to B. Gross and J. Harris on ovals of random plane curves publications.ias.edu/sarnak/paper/510 (2011) M. Shub, S. Smale, Complexity of Bezout s teorem II: volumes and probabilities, The collected papers of Stephen Smale, Volume 3 (pp )