Systems of polynomials equations Jean-Marc Azas Univ. of Toulouse - PowerPoint PPT Presentation

Random waves in Oxford june 2018 Systems of polynomials equations Jean-Marc Azaïs Univ. of Toulouse With D. Armentano. F.Dalmao and J. León 1

Random polynomials Polynomials with independent Gaussian coefficients were the first case to be extensively studied Kac(43), completely solved (var. CLT) in the 70’s Trigonometric polynomials of the form d X P d ( t ) = a j sin(2 π jt ) + b j cos(2 π jt ) j =1 have been studied by Granville and Wigman (2011) by Azaïs and Leon (2013) and Azaïs Dalmao and Leon (2016) ( without sine) 2

Polynomials systems The ensemble of Shub-Smale random polynomials was introduced in the early 90s by Kostlan Kostlan argues that this is the most natural distribution for a polynomial system. The exact expectation was obtained in the early 90's by geometric means, by Shub and Smale (1993). In 2004, 2005 Azaïs and Wschebor and Wschebor obtained by probabilistic methods the asymptotic variance as the number of equations and variables tends to infinity. Recently, Dalmao (2015) obtained the asymptotic variance and a CLT for the number of zeros as the degree d goes to infinity in the case of one equation in one variable 3

Framework We will first consider square systems with m equations and m variables and count the number of zeros . In a second step we will consider the rectangular case with more variables and measure volume of nodal sets. a ( ` ) X j t j , P ` ( t ) = | j | ≤ d j and t are vectors in R m 4

We assume ✓ d ◆ d ! ⇣ ⌘ a ( ` ) = = V ar j 1 ! . . . j m !( d − | j | )! . j j Adding a dummy variable to give homogeneity and restricting our attention to the unit sphere The numb. of zeros is the half of the numb. N of Zeros of the process with cov. h s, t i d on the unit sphere In that sense the KSS model is natural 5

Main result The degree d tends to infinity Theorem V ar ( N ) = V 2 0 < V 2 lim ∞ , ∞ < ∞ d m/ 2 d →∞ Interpretation: We have concentration : the mean is greater than the standard error. 6

Kac-Rice formula It is our main tool for expectation and variance Expectation Z S m E [ | det Y 0 ( t ) | | Y ( t ) = 0 ] E ( N ) = · p Y ( t ) (0) ds. Because of stat. on the Sphere, 1 the conditioning disappears 2 the integrand is constant giving E ( N ) = 2 d M/ 2 7

Variance V ar ( N ) = E ( N ( N − 1)) + E ( N ) − E 2 ( N ) Z ( S m ) 2 E [ | det Y 0 ( s ) det Y 0 ( t ) | | Y ( s ) = Y ( t ) = 0 ] E ( N ( N − 1)) = · p Y ( s ) , Y ( t ) (0 , 0) dsdt. We use invariance by rotation Z π Z sin( ψ ) m − 1 H (cos( ψ )) d ψ ( S m ) 2 H ( h s, t i ) ds dt = κ m κ m − 1 0 ✓ z ✓ z Z √ ◆ m − 1 d π ✓ ◆◆ = κ m κ m − 1 p sin p cos p H dz, d d d 0 8

Weak local limit of the projection after scaling No Global limit ! 9

1 d − m/ 2 V ar ( N ) = E ( N ( N − 1)) − ( E ( N )) 2 ⇤ ⇥ + 2 d m/ 2 ✓ z Z √ d π ◆ = 2 + κ m κ m − 1 sin m − 1 d ( m − 1) / 2 √ (2 π ) m d 0 σ 2 ( z ⇣ z ⇣ z d )  � √ ⇣ ⌘ ⌘⌘ − G (0 , 0) d )) m/ 2 G , D dz. ρ √ √ (1 − cos 2 d ( z d d √ We use a domination argument 10

The pointwise convergence is a direct consequence of the local limit For the domination we use A local part which is only uniformity of the convergence above + the fact that the variance kills the singularity of the density And a global part by difference with the independent case 11

Minorization At this stage nothing proves that the limit variance is positive ! Hermite decomposition Formally the number of zeros of Y can be represented by a Kac’s formula Z S m | det( Y 0 ( t )) | � ✏ ( Y ( t )) dt , N = lim ✏ ! 0 N ✏ with N ✏ := m 1 Y � ✏ ( y ) := 2 ✏ 1 {| y ` | < ✏ } ` = 1 12

The limit happens in both senses : a.s. and L2 The proof is easy because of the Bezout Th.: The number of points such that the N=u is bounded Proposition 1 N d : = N − 2 d m/ 2 ¯ X = I q,d , 2 d m/ 4 q =1 where I q,d = d m/ 4 Z X c γ H α ( Y ( t )) ¯ H β ( ¯ Y 0 ( t )) dt , 2 S m | γ | = q contains the formal coef. of the Dirac c γ and of the determinant 13

The variances of the different components in the different chaos add. It suffices to prove that the variance of the component in the second Chaos q=2 is positive . 14

Rectangular case When we have (m) more variables than equations (m’) The nodal set is an m-m’manifold (easy). The proof is very similar since we consider the same process We consider the m-m’ measure of the nodal set We have no factorial moment the integral is less degenerated 15

The Kac-Rice formula has a different form E [( V Y d ) 2 ] Z = S m ⇥ S m 1 1 2 (det( Y 0 2 | Y d ( t ) = Y d ( s ) = 0 ] d ( t ) T )) d ( s ) T )) E [(det( Y 0 d ( t ) Y 0 d ( s ) Y 0 × p Y d ( t ) , Y d ( s ) (0 , 0) dtds, 16

Central limit Theorem Federico Dalmao will present how Hermite decomposition permits to prove a CLT A contrario to the proof of the variance we have now to study the components in all the chaos !! 17

More general models The considered form of the covariance h s, t i d is a particular case of isotrope model on the sphere We can consider more general functions as g ( h s, t i ) There is a description of the eligible g (Kostlan 2001) 18

References • T. Letendre . Expected volume and Euler characteristic of random submanifolds. J. Funct. Anal. 270 (2016), no. 8, 3047-3110. T. Letendre and M. Puchol . Variance of the volume of random real algebraic submanifolds II. arXiv: Armentano, D., Azaïs, J. M., Dalmao, F., & León, J. R. (2018). On the asymptotic variance of the number of real roots of random polynomial systems. To appear in PAMS Armentano, D., Azaïs, J. M., Dalmao, F., & León, J. R . (2018).Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial system. Arxiv Armentano, D., Azaïs, J. M., Dalmao, F., & León, J. R. (2018) . Asymptotics for the Volume of the zero set for Kostlan-Shub-Smale polynomial systems. Working paper. 19

THANK-YOU 20