On the Problem of Separating Variables in Multivariate Polynomial Ideals

TL;DR

This work studies separating variables in multivariate polynomial ideals by introducing the algebra $A(I)$ of separated elements and developing algorithms to generate its elements. A reduction to the bivariate setting is achieved via the homomorphism $\phi$ that maps $x_i$ to $s x_i$ and $y_j$ to $t y_j$, enabling a precise analysis of principal ideals: $A(I)$ is simple when the generator $p$ is not in $K[X]$ or $K[Y]$, and a generator can be extracted from the corresponding pair $(F|_{s=1},G|_{t=1})$ of the bivariate problem. For dimension-zero ideals, $A(I)$ has finite codimension and can be computed by degree-bounded ansatz and Gröbner-basis reductions; combining a zero-dimensional ideal with a principal ideal yields finite generation and a feasible algorithm for $A(I)$. For arbitrary ideals, finiteness fails in general, so the paper offers two enumeration strategies—one based on degree-bounded linear algebra and another using Gröbner-bases and bivariate reduction—whose termination is guaranteed only when $A(I)$ is finitely generated. Overall, the results delineate when separated-variable polynomials can be algorithmically characterized and highlight open questions about guaranteed termination in the general case.

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.

Theorems & Definitions (32)

• Lemma 1
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• Lemma 2
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• Lemma 3
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• Lemma 4
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• Lemma 5
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