Toward a unified theory for common affine roots of general sets of multivariate polynomials

TL;DR

This paper proposes a unified framework for understanding common roots of multivariate polynomials, extending classical univariate theorems using error-correcting code methods and the footprint bound.

Contribution

It introduces a new perspective that replaces degree with leading monomial as the measure for multivariate polynomials, unifying factor and interpolation theorems.

Findings

Footprint bound provides sharp root count bounds for finite Cartesian sets.

Unified formulation of factor and interpolation theorems for multivariate polynomials.

Error-correcting codes techniques facilitate the extension of classical polynomial theorems.

Abstract

For univariate polynomials over arbitrary field the degree gives an upper bound on the number of roots (factor theorem) and as a related result for any finite point-set one can construct a polynomial of degree equal to the cardinality having all the points as roots (interpolation theorem). Tao noted in [48] that the theory of multivariate polynomials is not yet sufficiently matured to provide similar theorems with an equally simple relation between them. In the present paper we argue that for general multivariate polynomials the right measure for the size of the polynomial should not be the degree, but the leading monomial. In this setting the footprint bound [27] becomes a natural enhancement of the factor theorem providing a bound on the number of common roots of general multivariate polynomials which is sharp for all finite Cartesian product point-sets. As our main contribution, by using methods from the theory of error-correcting codes we establish a natural formulation of the interpolation theorem to the case of common roots of multivariate polynomials. In short the two theorems reduce to the same result, but for dual spaces, establishing the unification requested in [48]. We leave it for further research to possibly establish similar interpolation results taking one or more of the various concepts of multiplicity of multivariate polynomials into account.