Subresultants and the Shape Lemma

TL;DR

This paper investigates the relationship between resultants, subresultants, and the Shape Lemma in zero-dimensional ideals, providing new insights into when resultants generate elimination ideals and introducing a Poisson formula for resultants.

Contribution

It establishes conditions under which the resultant generates the elimination ideal and proves a Poisson formula for resultants from the hidden variable method.

Findings

Resultants can generate the elimination ideal in certain cases.

A Poisson formula for resultants is proven.

Connections between resultants, subresultants, and the Shape Lemma are clarified.

Abstract

In nice cases, a zero-dimensional complete intersection ideal over a field of characteristic zero has a Shape Lemma. There are also cases where the ideal is generated by the resultant and first subresultant polynomials of the generators. This paper explores the relation between these representations and studies when the resultant generates the elimination ideal. We also prove a Poisson formula for resultants arising from the hidden variable method.