[Paper Review] Symmetric semi-algebraic sets and non-negativity of symmetric polynomials

Cordian Riener|arXiv (Cornell University)|Sep 2, 2014

Polynomial and algebraic computation12 references4 citations

TL;DR

This paper generalizes Timofte's half-degree and degree principles for certifying non-negativity of symmetric polynomials by leveraging their representation in terms of power sum polynomials. It proves that for symmetric polynomials sparse in a subset of power sums, non-negativity over low-complexity symmetric varieties (with few distinct components) implies global non-negativity, enabling efficient optimization and decision procedures for symmetric semi-algebraic sets.

ABSTRACT

The question of how to certify the non-negativity of a polynomial function lies at the heart of Real Algebra and it also has important applications to Optimization. In the setting of symmetric polynomials Timofte provided a useful way of certifying non-negativity of symmetric polynomials that are of a fixed degree. In this note we present more general results which naturally generalize Timofte's setting. We investigate families of polynomials that allow special representations in terms of power-sum polynomials.These in particular also include the case of symmetric polynomials of fixed degree. Therefore, we recover the consequences of Timofte's original statements as a corollary. Thus, this note also provides an alternative and simple proof of Timofte's original statements.

Motivation & Objective

• To generalize Timofte's half-degree and degree principles for symmetric polynomials using power sum polynomial representations.
• To characterize symmetric polynomials that admit sparse representations in terms of a subset of power sums.
• To provide an elementary proof of non-negativity certification via power sum structure, avoiding advanced algebraic geometry.
• To establish that non-negativity on low-complexity symmetric varieties (with ≤d distinct components) implies global non-negativity for symmetric polynomials of degree 2d.
• To extend these results to copositivity and feasibility of symmetric semi-algebraic sets using topological and algebraic arguments.

Proposed method

• Represents symmetric polynomials via the fundamental theorem of symmetric polynomials using power sum generators $ p_i = \sum X_j^i $.
• Introduces the concept of $ J $-sparsity, where a symmetric polynomial depends only on a subset $ J \subset \{1,\dots,n\} $ of power sums.
• Uses Lagrange multipliers on symmetric level sets $ H(a_1,\dots,a_k) $ to analyze minimizers and show they must lie in the set $ A_k $ of points with at most $ k $ distinct components.
• Applies a local perturbation argument using Lemma 3.2 to show that minimizers with more than $ k $ distinct components cannot exist under non-degeneracy conditions.
• Reduces the non-negativity certification problem to checking non-negativity on a finite union of lower-dimensional symmetric varieties parameterized by $ k $-partitions of $ n $.
• Extends results to copositivity by considering $ f(X_1^2,\dots,X_n^2) $, which depends only on even-powered symmetric sums.

Experimental results

Research questions

• RQ1 Can Timofte's half-degree principle be generalized beyond full power sum bases to sparse subsets of power sums?
• RQ2 Under what conditions does non-negativity of a symmetric polynomial on varieties with at most $ d $ distinct components imply global non-negativity?
• RQ3 How does the structure of power sum polynomials enable efficient certification of non-negativity in symmetric optimization problems?
• RQ4 Is the non-negativity of a symmetric polynomial determined by its behavior on symmetric varieties of bounded component count, and how does this relate to sparsity in power sum generators?
• RQ5 Does the choice of power sum generators affect the validity of the degree and half-degree principles, and can alternative bases yield similar results?

Key findings

• A symmetric polynomial of degree $ 2d $ is non-negative on $ \mathbb{R}^n $ if and only if it is non-negative on all points with at most $ d $ distinct components, generalizing Timofte's half-degree principle.
• For symmetric polynomials $ J $-sparse in a set $ J $ of power sums, non-negativity on the corresponding symmetric varieties implies global non-negativity, reducing complexity to polynomial dependence on $ n $.
• The minimizer set of a symmetric polynomial on a symmetric level set $ H(a_1,\dots,a_k) $ must intersect $ A_k $, the set of points with at most $ k $ distinct components, under mild conditions.
• If a symmetric polynomial $ f $ is non-negative on all points with at most $ k $ distinct components, then $ f $ is globally non-negative, even when $ f $ is not a full symmetric polynomial.
• The results extend to copositivity: if $ f(X_1^2,\dots,X_n^2) $ is non-negative, then $ f $ is copositive, and this can be certified via low-complexity symmetric varieties.
• The proof relies on the structure of power sum polynomials and shows that non-degenerate minimizers must lie in low-complexity symmetric loci, enabling algorithmic certification.