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

Systems of polynomials equations Jean-Marc Azaïs Univ. of Toulouse With D. Armentano. F.Dalmao and J. León

Random waves in Oxford june 2018

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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

Pd(t) =

d

X

j=1

aj sin(2πjt) + bj cos(2πjt)

have been studied by Granville and Wigman (2011) by Azaïs and Leon (2013) and Azaïs Dalmao and Leon (2016) ( without sine)

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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

• ne equation in one variable

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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.

P`(t) = X

|j|≤d

a(`)

j tj,

j and t are vectors in Rm

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We assume

V ar ⇣ a(`)

j

⌘ = ✓d j ◆ = d! j1! . . . jm!(d − |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.

hs, tid

• n the unit sphere

In that sense the KSS model is natural

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Main result

lim

d→∞

V ar(N) dm/2 = V 2

∞,

0 < V 2

∞ < ∞ Theorem Interpretation: We have concentration: the mean is greater than the standard error.

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The degree d tends to infinity

Kac-Rice formula

It is our main tool for expectation and variance

Expectation

Because of stat. on the Sphere, 1 the conditioning disappears 2 the integrand is constant giving

E(N) = Z

Sm E[| det Y0(t)| |Y(t) = 0]

·pY(t)(0)ds.

E(N) = 2dM/2

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E(N(N − 1)) = Z

(Sm)2 E[| det Y0(s) det Y0(t)| |Y(s) = Y(t) = 0]

·pY(s),Y(t)(0, 0)dsdt.

Variance

V ar(N) = E(N(N − 1)) + E(N) − E2(N)

We use invariance by rotation Z

(Sm)2 H(hs, ti) ds dt = κmκm−1

Z π sin(ψ)m−1H(cos(ψ)) dψ = κmκm−1 p d Z √

dπ

sin ✓ z p d ◆m−1 H ✓ cos ✓ z p d ◆◆ dz,

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

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We use a domination argument

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d−m/2V ar (N) = 1 dm/2 ⇥ E(N(N − 1)) − (E(N))2⇤ + 2 = 2 + κmκm−1 (2π)m Z √

dπ

sinm−1 ✓ z √ d ◆ d(m−1)/2  σ2( z

√ d)

(1 − cos2d( z

√ d))m/2 G

⇣ ρ ⇣ z √ d ⌘ , D ⇣ z √ d ⌘⌘ − G(0, 0)

• dz.

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

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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

N = lim

✏!0 N✏ with N✏ :=

Z

Sm | det(Y0(t))| ✏(Y(t))dt,

✏(y) :=

m

Y

`=1

1 2✏1{|y`|<✏}

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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

¯ Nd : = N − 2dm/2 2dm/4 =

1

X

q=1

Iq,d, where Iq,d = dm/4 2 Z

Sm

X

|γ|=q

cγHα(Y(t)) ¯ Hβ( ¯ Y0(t))dt,

contains the formal coef. of the Dirac and of the determinant cγ

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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.

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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

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E[(VYd)2] = Z

Sm⇥Sm

E[(det(Y0

d(t)Y0 d(t)T))

1 2 (det(Y0

d(s)Y0 d(s)T))

1 2 |Yd(t) = Yd(s) = 0]

× pYd(t),Yd(s)(0, 0)dtds,

The Kac-Rice formula has a different form

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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 !!

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More general models The considered form of the covariance

hs, tid

is a particular case of isotrope model on the sphere We can consider more general functions as g(hs, ti)

There is a description of the eligible g (Kostlan 2001)

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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.

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THANK-YOU

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