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

Olav Geil

TL;DR

The paper addresses bounding the number of common affine roots of general multivariate polynomials by promoting the leading monomial as the size measure and employing the footprint bound $Δ_{ prec}(I)$ to obtain sharp bounds on common roots for finite point-sets. It establishes a dual, interpolation-like result in the dual space, linking problems (I) and (II) through a Feng–Rao/Forney framework that extends over arbitrary fields and recovers univariate degree-based bounds as special cases. Central to the approach is translating root-count questions into linear-algebraic questions via relative generalized Hamming weights, yielding tight bounds and a unified view across polynomial and coding-theoretic settings. The work also highlights future directions toward multiplicity-aware bounds and broader algebraic structures, with potential implications for algebraic-geometric codes and secret-sharing schemes.

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 [44] 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 [25] 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 [44]. 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.

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

TL;DR

The paper addresses bounding the number of common affine roots of general multivariate polynomials by promoting the leading monomial as the size measure and employing the footprint bound to obtain sharp bounds on common roots for finite point-sets. It establishes a dual, interpolation-like result in the dual space, linking problems (I) and (II) through a Feng–Rao/Forney framework that extends over arbitrary fields and recovers univariate degree-based bounds as special cases. Central to the approach is translating root-count questions into linear-algebraic questions via relative generalized Hamming weights, yielding tight bounds and a unified view across polynomial and coding-theoretic settings. The work also highlights future directions toward multiplicity-aware bounds and broader algebraic structures, with potential implications for algebraic-geometric codes and secret-sharing schemes.

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 [44] 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 [25] 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 [44]. 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.
Paper Structure (5 sections, 9 theorems, 43 equations)

This paper contains 5 sections, 9 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Main results on multivariate polynomials
  3. Results in linear algebra setting
  4. Proof of Theorem
  5. Concluding remarks

Key Result

Proposition 1

Let ${\mathbb{F}}$ be a field, let $m \geq 1$ be an integer, and $d \geq 0$. If $A \subseteq {\mathbb{F}}^m$ has cardinality less than ${d+m}\choose{m}$, then there exists a non-zero polynomial $F \in {\mathbb{F}}[X_1, \ldots , X_n]$ of degree at most $d$ having $A$ as roots.

Theorems & Definitions (25)

  • Proposition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Theorem 5
  • proof
  • Remark 6
  • Remark 7
  • Theorem 8
  • proof
  • ...and 15 more