On Computing the Elimination Ideal Using Resultants with Applications to Gr\"obner Bases

TL;DR

This paper explores the relationship between resultants and Gr"obner bases in polynomial elimination, focusing on the bivariate case and analyzing multiplicity differences in elimination ideals.

Contribution

It provides new insights into the connection between resultants and elimination ideals, especially in the bivariate case, using elementary methods.

Findings

Analyzes the relation between the variety of the resultant and the ideal they generate.

Studies the principal elimination ideal in the bivariate case.

Examines the multiplicity differences between factors of the generator and the resultant.

Abstract

Resultants and Gr\"obner bases are crucial tools in studying polynomial elimination theory. We investigate relations between the variety of the resultant of two polynomials and the variety of the ideal they generate. Then we focus on the bivariate case, in which the elimination ideal is principal. We study - by means of elementary tools - the difference between the multiplicity of the factors of the generator of the elimination ideal and the multiplicity of the factors of the resultant.