Average Number of Roots of Systems of Equations

Abstract

Let V₁, …, V_n be finite-dimensional spaces of smooth functions on a smooth n-manifold X. We determine a relationship between the average number of solutions to systems of equations {f_i = a_i ∣ f_i ∈ V_i, a_i ∈ ℝ, i =1, …, n} and mixed volumes of convex bodies. To this end, assuming the spaces V_i to be normed, we construct (1) measures on the spaces of systems of equations and (2) Banach convex bodies in X, i.e., families of centrally symmetric convex bodies in fibers of the cotangent bundle of X. We define the volume of a Banach convex body as the symplectic volume of the union of the corresponding family. It turns out that the average number of solutions equals the mixed symplectic volume of the Banach convex bodies corresponding to the spaces V_i. Moreover, on the right-hand side of the equality arbitrary smooth strictly convex Banach bodies can appear. Previously the case of Euclidean spaces V_i have been considered. In this case, the Banach bodies are families of ellipsoids.