A Generalization of Habicht's Theorem for Subresultants of Several Univariate Polynomials

TL;DR

This paper extends Habicht's classical theorem on subresultants from two univariate polynomials to multiple polynomials, enhancing understanding and computational methods in algebraic geometry.

Contribution

It provides the first generalization of Habicht's theorem for subresultants of several univariate polynomials, a challenging extension in algebraic computation.

Findings

Generalized Habicht's theorem to multiple polynomials

Improved theoretical understanding of multivariate subresultants

Potential for more efficient algebraic computations

Abstract

Subresultants of two univariate polynomials are one of the most classic and ubiquitous objects in computational algebra and algebraic geometry. In 1948, Habicht discovered and proved interesting relationships among subresultants. Those relationships were found to be useful for both structural understanding and efficient computation. Often one needs to consider several (possibly more than two) polynomials. It is rather straightforward to generalize the notion of subresultants to several polynomials. However, it is not obvious (in fact, quite challenging) to generalize the Habicht's result to several polynomials. The main contribution of this paper is to provide such a generalization.