On Newton's identities in positive characteristic

TL;DR

This work analyzes Newton's identities in positive characteristic and shows how elementary symmetric polynomials $e_1,\dots,e_n$ can still be recovered as rational functions in power sums $p_i$ via an explicit algorithm, despite obstructions to generating the symmetric polynomial ring from power sums in characteristic $p$. It proves the foundational result $Q(R[e_1,\dots,e_n]) = Q(R[p_1,p_2,\dots])$ for any commutative ring $R$, and then thoroughly studies the structure of the subalgebra generated by power sums in positive characteristic, demonstrating non-finite generation when $n \ge r$ and detailing a containment $K[p_\infty] \subset E \subsetneq K[e_1,\dots,e_n]$. The paper also extends to Fr-algebras, provides evaluation criteria for the resulting rational expressions, and connects valuations in discrete valuation rings to the behavior of power sums, offering a broader algebraic framework for symmetric polynomials in positive characteristic. These results yield both theoretical insight and practical algorithms for recovering roots and symmetric data from power sums in settings where characteristic interferes with classical Newton identities.

Abstract

Newton's identities provide a way to express elementary symmetric polynomials in terms of power polynomials over fields of characteristic zero. In this article, we study the failure of this relation in positive characteristic and what can be recovered. In particular, we show how one can write the elementary symmetric polynomials as rational functions in the power polynomials over any commutative unital ring.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Theorem \ref{['mainthm']}
• Theorem 2.1: Fundamental Theorem on Symmetric Polynomials
• proof
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• proof