# Generalized characteristic polynomials

## Abstract

We generalize the notion of characteristic polynomial for a system of linear equations to systems of multivariate polynomial equations. The generalization is natural in the sense that it reduces to the usual definition when all the polynomials are linear. Whereas the constant coefficient of the characteristic polynomial of a linear system is the determinant, the constant coefficient of the general characteristic polynomial is the *resultant* of the system. This construction is applied is solve a traditional problem with efficient methods for solving systems of polynomial equations: the presence of infinitely many solutions “at infinity”. We give a single-exponential time method for finding all the isolated solution points of a system of polynomials, even in the presence of infinitely many solutions at infinity or elsewhere.

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