Statistical algebraic geometry

Background: In statistical algebraic geometry, we put a Gaussian probability measure on the space of polynomials of degree $N$ in $m$ real or complex variables. For simplicity, we think mainly of the $U(m + 1)$-invariant Gaussian measure in the complex case and the $O(m + 1)$ invariant measure in the real case. We then consider probabilities and expected values for interesting random variables. The real and complex cases are quite different, since deterministic problems in the complex case can become random in the real case.

Questions for real algebraic plane curves:

• Consider the ensemble of plane algebraic curves of degree $N$. Let the random variable be: the number of components of the curve. What is the most probable number of connected components? What is the expected number?

• Consider random spherical harmonics of degree $N$. Let the random variable be the number of nodal domains (i.e. components of the complement of the zero set of the harmonic). What is the most probable number of nodal domains? The expected number?

Random real fewnomials. We fix a number $f$. In dimension $m$, we select $m$ real fewnomials of degree $N$ at random, each with at most $f$ monomials. We pick the spectrum of each fewnomial at random ($f$ lattice points in $\mathbb{Z}_+^m \cap N \Sigma$). We then pick the coefficients of these fewnomials at random from the $O(m + 1)$ ensemble. The problem is:

Question: What is the expected number of real zeros of a random fewnomial system of degree $N$ with $f$ monomials in each fewnomial?

The current bound, due to Khovanski, is

$$\char93 \mbox{real zeros}\;\; \leq 2^m 2^{f (f - 1)/2} (m + 1)^f.$$

It is believed to be an enormous over-estimate.

Zeros of random real fewnomials with fixed Newton polytope. We now pick $m$ random fewnomials $p_1, \dots, p_m$ with prescribed Newton polytopes $\Delta_1, \dots, \Delta_m$ and fixed fewnomial number $f$. How does the number of simultaneous zeros behave as the polytopes are dilated, $\Delta_j \to N \Delta_j$. I.e. we increase the degrees, but keep the fewnomial number $f$ fixed and keep the spectra in the dilates of the polytopes.

Zeros of random real Kac fewnomials. We ask the same questions but define random real fewnomial as $\sum_{\alpha} c_{\alpha} x^{\alpha}$ where $c_{\alpha}$ are normal. That is, we do not use projective space to define norms of monomials. [The number of real zeros then goes way down.]

Current result: Shiffman and I currently have an exact formula for the expected number of real zeros of random fewnomial ensembles, but we have not yet found its asymptotics.

Proposition. The density $K_f^{N}(x)$ of real zeros of random $f$-fewnomial systems of degree $N$ is given by:

$$\begin{array}{l} K_f^{N}(x) = \frac{1}{\pi^{knm}} \frac{Q}{\v... ...{x = y} }}{[{\sqrt{\Pi_{N\vert S}(x,x)}}]^m }, \;\; \end{array}$$

where $Q := \int_{\mathbb{R}^m} \vert\xi\vert \exp\left( -{\langle \xi,\xi\rangle}\right) d \xi,$ and where $\Pi_{N\vert S}$ is the Szegö kernel for the spectrum $S$

$$\Pi_{N \vert S}(x, y) = \sum_{\beta \in S} \; \binom{N}{\beta} x^{\beta} y^{\beta} .$$

Here, ${\mathcal S}(N, f)$ is the set of possible spectra.

Critical points of holomorphic sections. Critical points of holomorphic functions have a long history (Picard-Lefschetz, Milnor-Orlik, Arnold, etc.). Critical points of holomorphic sections have arisen recently in string theory, where they are `supersymmetric vacua'. Mike Douglas has posed the problem of counting them, finding how they are distributed, and many other statistic problems relevant in string/M theory.

Critical points of a holomorphic section $s\in H^0(M, L)$ of a holomorphic line bundle depend on a choice of Hermitian metric $h$ or connection $\nabla$, which we usually pick to be the metric connection. The equation reads $\nabla s (z) = 0$ and hence the number of critical points depends on the connection or metric.

(contributed by Steve Zelditch)

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