[Paper Review] On the theory of symmetric polynomials

M. Г. Крейн|arXiv (Cornell University)|May 9, 2014

Mathematics and Applications3 citations

TL;DR

This paper presents an algebraic approach to counting roots of symmetric polynomials inside the unit circle |x|=1 using quadratic forms derived from power sums of roots. By constructing a Hermitian form from symmetric polynomial coefficients, the method algebraically establishes Cohn’s theorem and its generalization, enabling exact root localization without analytic continuation or Rouché’s theorem.

ABSTRACT

A translation from Russian of the famous paper by Mark Krein "On the theory of symmetric polynomials" published in Math. Sb., vol. 40, no. 3, 1933, pp. 271-283.

Motivation & Objective

• To provide an algebraic alternative to Cohn’s root-counting theorem for symmetric polynomials, which traditionally relies on Rouché’s theorem and continuity arguments.
• To establish a criterion for the number of roots of a symmetric polynomial inside the unit circle |x|=1 using a Hermitian form constructed from power sums of roots.
• To generalize Cohn’s theorem using algebraic techniques, avoiding analytic methods such as continuity or residue calculus.
• To draw analogies between symmetric and real polynomials, particularly in root interlacing and sign-definite forms.
• To offer a constructive method for determining root distribution (inside/outside |x|=1) via the signature of a quadratic form derived from symmetric polynomial coefficients.

Proposed method

• Define the Hermitian form $\mathfrak{S} = \sum_{i,k=0}^{n-1} s_{i-k} x_i \overline{x}_k$, where $s_k$ are power sums of roots of a symmetric polynomial $g(x)$.
• Express $s_k$ in terms of roots on the unit circle ($\varepsilon_s$) and symmetric pairs ($\beta_t, \beta_t^*$), leading to a decomposition of $\mathfrak{S}$ into sum of squared moduli.
• Use the signature of $\mathfrak{S}$ (number of positive/negative squared terms) to determine root distribution: $\pi - \nu$ roots on |x|=1, $\nu$ symmetric pairs outside/in.
• Relate $\mathfrak{S}$ to the Schur-Cohn form $\mathfrak{H}$ to derive Cohn’s theorem algebraically, bypassing Rouché’s theorem.
• Establish analogies with real polynomials by transforming symmetric polynomials via $f(x) = (x-i)^n g\left(\frac{x+i}{x-i}\right)$, linking symmetric and real root distributions.
• Use the form $\left[\frac{F(x)f(y)-F(y)f(x)}{x-y}\right]_n$ to characterize interlacing roots, analogous to Hurwitz’s criterion.

Experimental results

Research questions

• RQ1 Can Cohn’s theorem on root distribution of symmetric polynomials be proven algebraically without relying on complex analysis or continuity arguments?
• RQ2 What is the relationship between the signature of a quadratic form built from power sums of roots and the location of roots relative to the unit circle?
• RQ3 How can the theory of symmetric polynomials be systematically related to that of real polynomials via algebraic transformations?
• RQ4 Under what conditions do two symmetric polynomials have interlacing roots on the unit circle, and how can this be characterized algebraically?
• RQ5 What algebraic criterion determines whether a symmetric polynomial has all roots on the unit circle or in symmetric pairs outside/in?

Key findings

• The Hermitian form $\mathfrak{S} = \sum_{i,k=0}^{n-1} s_{i-k} x_i \overline{x}_k$ has signature $\pi - \nu$ for roots on |x|=1 and $\nu$ symmetric pairs outside/in, with kernel dimension $d = n - \pi - \nu$.
• The greatest common divisor $D(x)$ of $g(x)$ and $g^*(x)$ has degree $d$, and $g(x)/D(x)$ has $\pi - \nu$ roots on |x|=1 and $\nu$ symmetric pairs outside/in.
• Cohn’s theorem is recovered algebraically: the number of roots of a symmetric polynomial inside |x|=1 equals the number of roots of its derivative outside |x|=1.
• The form $\mathfrak{S}$ decomposes into sum of squared moduli of linear forms in $x_i$, corresponding to roots on the unit circle and symmetric pairs.
• Analogues of classical results for real polynomials (e.g., Hermite’s criterion, interlacing conditions) are established for symmetric polynomials via algebraic forms.
• The transformation $f(x) = (x-i)^n g\left(\frac{x+i}{x-i}\right)$ maps symmetric polynomials to real polynomials, enabling transfer of root distribution theorems.