Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Real Gaussian exponential sums via a real moment map

Léo Mathis

TL;DR

This work analyzes the expected number of real zeros of Gaussian exponential sums by embedding the problem in Gaussian random fields and employing Adler–Taylor theory. A real moment map mu, derived from a convex potential Phi, turns the zero-count into a Riemannian volume on the Newton polytope, yielding a real BKK-type result. The main contributions include a local monotonicity analysis showing that interior additions can decrease the zero count in some regions (negating a local Bürgisser conjecture) and a global unbounded monotonicity phenomenon for distant additions, plus new lower bounds for Aronszajn multiplication. The methodology unifies convex geometry, Kac–Rice theory, and Veronese-type embeddings to connect the geometry of supports with zero statistics, and extends to a complex setting that recovers the classical BKK theorem in the appropriate limit. The results provide new insights into how sparse real/complex exponential sums behave under perturbations and combinatorial operations, with potential applications to sparse polynomial systems and random field theory.

Abstract

We study the expected number of solutions of a system of identically distributed exponential sums with centered Gaussian coefficient and arbitrary variance. We use the Adler and Taylor theory of Gaussian random fields to identify a moment map which allows to express the expected number of solution as an integral over the Newton polytope, in analogy with the Bernstein Khovanskii Kushnirenko Theorem. We apply this result to study the monotonicity of the expected number of solution with respect to the support of the exponential sum in an open set. We find that, when a point is added in the support in the interior of the Newton polytope there exists an open sets where the expected number of solutions decreases, answering negatively to a local version of a conjecture by Bürgisser. When the point added in the support is far enough away from the Newton polytope we show that there is an unbounded open set where the number of solution decreases. We also prove some new lower bounds for the Aronszajn multiplication of exponential sums.

Real Gaussian exponential sums via a real moment map

TL;DR

This work analyzes the expected number of real zeros of Gaussian exponential sums by embedding the problem in Gaussian random fields and employing Adler–Taylor theory. A real moment map mu, derived from a convex potential Phi, turns the zero-count into a Riemannian volume on the Newton polytope, yielding a real BKK-type result. The main contributions include a local monotonicity analysis showing that interior additions can decrease the zero count in some regions (negating a local Bürgisser conjecture) and a global unbounded monotonicity phenomenon for distant additions, plus new lower bounds for Aronszajn multiplication. The methodology unifies convex geometry, Kac–Rice theory, and Veronese-type embeddings to connect the geometry of supports with zero statistics, and extends to a complex setting that recovers the classical BKK theorem in the appropriate limit. The results provide new insights into how sparse real/complex exponential sums behave under perturbations and combinatorial operations, with potential applications to sparse polynomial systems and random field theory.

Abstract

We study the expected number of solutions of a system of identically distributed exponential sums with centered Gaussian coefficient and arbitrary variance. We use the Adler and Taylor theory of Gaussian random fields to identify a moment map which allows to express the expected number of solution as an integral over the Newton polytope, in analogy with the Bernstein Khovanskii Kushnirenko Theorem. We apply this result to study the monotonicity of the expected number of solution with respect to the support of the exponential sum in an open set. We find that, when a point is added in the support in the interior of the Newton polytope there exists an open sets where the expected number of solutions decreases, answering negatively to a local version of a conjecture by Bürgisser. When the point added in the support is far enough away from the Newton polytope we show that there is an unbounded open set where the number of solution decreases. We also prove some new lower bounds for the Aronszajn multiplication of exponential sums.

Paper Structure

This paper contains 21 sections, 63 theorems, 244 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Methods
  4. Other results
  5. Preliminaries
  6. Convex geometry
  7. Legendre transform
  8. Quadratic forms and Ellipsoids
  9. Gaussian vectors
  10. The Kac-Rice formula with zonoids
  11. GRFs and Adler-Taylor theory
  12. General theory
  13. Local monotonicity of GRFs
  14. Gaussian Exponential sums
  15. Random polynomials
  16. ...and 6 more sections

Key Result

Theorem 1

Let $A\subset \mathbb{Z}^m$ be a finite subset and for all $i=1,\cdots, m$, let $Q^{i}(w):=\sum_{a\in A}c^i_a w^a$ with $c_a^i\in \mathbb{C}$. Then for a generic choice of $c_a^i$, we have where $P:=\mathop{\mathrm{conv}}\nolimits(A)\subset \mathbb{R}^m$ is the convex hull of $A$, also called the Newton polytope.

Figures (2)

  • Figure 1: The ellipsoids of $\mathcal{X}$ and $\mathcal{X}_0$
  • Figure 2: The set $U_-$ (yellow) in the Newton Polytope (red) for $d=1$ and different cases of $a_0$ (blue)

Theorems & Definitions (149)

  • Theorem : BKK
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 139 more