Symmetric nonnegative functions, the tropical Vandermonde cell and superdominance of power sums

TL;DR

This paper investigates the structure of symmetric nonnegative and sums of squares functions of fixed degree, exploring their limit cones, tropicalizations, and the superdominance order on partitions, revealing new insights into their geometric and combinatorial properties.

Contribution

It introduces the concept of limit cones for symmetric nonnegative and sums of squares functions, and connects tropical dual cones with the superdominance order on partitions.

Findings

Explicit examples of nonnegative polynomials not expressible as sums of squares.

Tropicalizations of dual cones are characterized by superdominance order.

Power sum functions follow the superdominance partial order.

Abstract

We study nonnegative and sums of squares symmetric (and even symmetric) functions of fixed degree. We can think of these as limit cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree as the number of variables goes to infinity. We compare these cones, including finding explicit examples of nonnegative polynomials which are not sums of squares for any sufficiently large number of variables, and compute the tropicalizations of their dual cones in the even symmetric case. We find that the tropicalization of the dual cones is naturally understood in terms of the overlooked superdominance order on partitions. The power sum symmetric functions obey this same partial order (analogously to how term-normalized power sums obey the dominance order).