On the Problem of Separating Variables in Multivariate Polynomial Ideals

TL;DR

This paper develops algorithms to identify elements in multivariate polynomial ideals that can be separated into parts involving only x-variables and only y-variables, focusing on principal and zero-dimensional ideals.

Contribution

It introduces finite-step algorithms for separating variables in principal and zero-dimensional polynomial ideals within multivariate polynomial rings.

Findings

Algorithms successfully compute all separable elements in specified ideals.

Method applies to principal and zero-dimensional ideals.

Provides a computational approach for variable separation in polynomial ideals.

Abstract

For a given ideal I in K[x_1,...,x_n,y_1,...,y_m] in a polynomial ring with n+m variables, we want to find all elements that can be written as f-g for some f in K[x_1,...,x_n] and some g in K[y_1,...,y_m], i.e., all elements of I that contain no term involving at the same time one of the x_1,...,x_n and one of the y_1,...,y_m. For principal ideals and for ideals of dimension zero, we give a algorithms that compute all these polynomials in a finite number of steps.