The Differential Counting Polynomial

TL;DR

This paper introduces the differential counting polynomial, a new tool for quantitatively analyzing the solution sets of polynomial nonlinear differential equations, helping determine completeness of solutions.

Contribution

It presents the differential counting polynomial, generalizing existing polynomials, to assess the solution set completeness for differential equations.

Findings

Differential counting polynomial generalizes dimension and algebraic counting polynomials.

Under mild assumptions, it determines if a solution set is complete.

Applicable to both ordinary and partial differential equations.

Abstract

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common generalization of the dimension polynomial and the (algebraic) counting polynomial. Under mild additional asumptions, the differential counting polynomial decides whether a given set of solutions of a system of differential equations is the complete set of solutions.