Zero sets and factorization of polynomials of two variables

TL;DR

This paper explores the relationship between zero sets and factorization of bivariate polynomials with real coefficients, providing criteria for common factors and illustrating differences in quaternionic cases.

Contribution

It generalizes the zero-factor relationship to two-variable polynomials and introduces criteria for common factors based on zero sets, including quaternionic cases.

Findings

Criteria for when two polynomials share a common factor based on zero sets

Existence of polynomials with the same zero set but no common factor in quaternionic coefficients

Extension of classical polynomial factorization concepts to multivariable and quaternionic contexts

Abstract

The relationship between a polynomial's zeros and factors is well known. If a is a zero of f(x) then (x-a) is a factor of f(x). In this paper, we generalize this idea to polynomials of two variables and with real coefficients. We consider the zero sets of two variable polynomials and give criterion to when two polynomials with the same zero set have a common factor with the same zero set. When the coefficients of the polynomials are not in a field, but the division algebra of Quaternions, we provide an example of two polynomials with the same zero set and no common factor.