The multivariate Hermite method for counting real and complex solutions to polynomial systems

TL;DR

This paper introduces the multivariate Hermite criterion, an effective algorithm for counting real and complex solutions of polynomial systems, with an implementation in Macaulay2.

Contribution

It presents a new multivariate Hermite method for solution counting, extending previous techniques to any number of variables with practical implementation.

Findings

Provides a practical algorithm for solution counting in polynomial systems.

Includes an open-source Macaulay2 implementation.

Applicable to systems with any finite number of variables.

Abstract

This note presents the multivariate Hermite criterion: a practical and powerful algorithm for determining the number of distinct real and complex roots of a zero-dimensional system of polynomials in any finite number of variables. The final section includes an implementation in Macaulay2, a free and open-source computer algebra system.