Average number of solutions

TL;DR

This paper establishes a relationship between the average number of solutions of certain systems of equations on manifolds and mixed volumes of convex bodies, extending previous Euclidean results to Banach metrics.

Contribution

It generalizes the connection between solution counts and mixed volumes from Euclidean to Banach metrics on manifolds.

Findings

Average number of solutions equals mixed symplectic volume of Banach convex bodies.

Constructs measure in the space of systems using Banach metrics.

Extends earlier Euclidean results to more general Banach settings.

Abstract

Let $X$ be an $n$-dimensional manifold and $V_1,\ldots,V_n\subset C^\infty(X,\mathbb R)$ finite-dimensional vector spaces. For systems of equations $\{f_i = a_i\colon\: f_i\in V_i,\:a_i \in\mathbb R,\:i=1,\ldots,n\}$ we discover a relationship between the average number of their solutions and mixed volumes of convex bodies. To do this, we choose Banach metrics in the spaces $V_i$. Using these metrics, we construct 1) the measure in the space of systems, and 2) Banach convex bodies in $X$, i.e., collections of centrally symmetric convex bodies in the fibers of the cotangent bundle of $X$. It turns out that the average number of solutions is equal to the mixed symplectic volume of Banach convex bodies. Earlier this result was obtained for Euclidean metrics in spaces $V_i$. In Euclidean case, the Banach convex bodies are the collections of ellipsoids.