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Polynomial maps and polynomial sequences in groups

Ya-Qing Hu

Abstract

This paper develops a theory of polynomial maps from commutative semigroups to arbitrary groups and proves that it has desirable formal properties when the target group is locally nilpotent. We apply this theory to solve Waring's Problem for Heisenberg groups in a sequel to this paper.

Polynomial maps and polynomial sequences in groups

Abstract

This paper develops a theory of polynomial maps from commutative semigroups to arbitrary groups and proves that it has desirable formal properties when the target group is locally nilpotent. We apply this theory to solve Waring's Problem for Heisenberg groups in a sequel to this paper.

Paper Structure

This paper contains 11 sections, 53 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Work of Leibman
  4. Our generalization
  5. Main results
  6. Polynomial Maps
  7. Continuous Polynomial Maps
  8. Estimation of the Degree
  9. Polynomial Sequences
  10. Symmetric Polynomial Maps
  11. Polynomial Sets

Key Result

Theorem 1

Let $S$ be any nonempty commutative semigroup, $G$ be any group and $f,f':S\rightarrow G$ be polynomial maps of degree $\le d$ and respectively $\le d'$. If the subgroup $\langle f,f'\rangle$ generated by $f(S)$ and $f'(S)$ is nilpotent, then the (elementwise) product is a polynomial map.

Theorems & Definitions (131)

  • Theorem 1
  • Remark
  • Lemma 1
  • proof
  • Definition 1
  • Remark
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 121 more