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Linear equations and multiplicative polynomial equations in infinitely many variables

Melvyn B. Nathanson, David A. Ross

TL;DR

The paper investigates when solvability of every finite subset of an infinite collection of polynomial equations in infinitely many variables implies solvability of the entire infinite system. It extends the classical finite-implies-infinite paradigm from linear equations to a structured class of multiplicative polynomials by introducing coefficients in $\ell^{q/(q-d)}$ and proving exact and approximate finite-subset-to-infinite solvability results. The approach combines compactness arguments on $\ell^q$-balls, continuity of multiplicative components, and the finite-intersection property, with refinements allowing coordinatewise bounds. An application to Dirichlet series demonstrates interpolation of prescribed values via Dirichlet-type sums within this framework. The paper concludes with open questions about extending these results to broader polynomial classes and subfields of $\mathbb C$.

Abstract

This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a solution for the infinite set of equations.

Linear equations and multiplicative polynomial equations in infinitely many variables

TL;DR

The paper investigates when solvability of every finite subset of an infinite collection of polynomial equations in infinitely many variables implies solvability of the entire infinite system. It extends the classical finite-implies-infinite paradigm from linear equations to a structured class of multiplicative polynomials by introducing coefficients in and proving exact and approximate finite-subset-to-infinite solvability results. The approach combines compactness arguments on -balls, continuity of multiplicative components, and the finite-intersection property, with refinements allowing coordinatewise bounds. An application to Dirichlet series demonstrates interpolation of prescribed values via Dirichlet-type sums within this framework. The paper concludes with open questions about extending these results to broader polynomial classes and subfields of .

Abstract

This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a solution for the infinite set of equations.
Paper Structure (7 sections, 9 theorems, 123 equations)

This paper contains 7 sections, 9 theorems, 123 equations.

Table of Contents

  1. Finitely many implies infinite many
  2. Polynomials in infinitely many variables
  3. Multiplicative polynomials
  4. Approximate finite implies exact infinite
  5. An application to Dirichlet series
  6. Open problems
  7. The theorems of F. Riesz and Abian-Eslami

Key Result

Theorem 1

Let $(p,q)$ be a conjugate pair with $p>1$ and $q>1$ and let $\mathbf a_i = \left( a_{i,j} \right)_{j=1}^{\infty} \in \ell^p$ for all $i \in \mathbf N$. Consider the linear equation in infinitely many variables Let $M > 0$. If, for every finite subset $S$ of $\mathbf N$, there is a sequence $\mathbf x_S \in \ell^q$ such that $\| \mathbf x_S\|_q \leq M$ and $L_i( \mathbf x_S) = b_i$ for all $i \

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof
  • Theorem 5
  • ...and 4 more