Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Power mean inequalities and sums of squares (2303.11823v1)

Published 21 Mar 2023 in math.AG

Abstract: For fixed degree and increasing number of variables the dimension of the vector space of $n$-variate real symmetric homogeneous polynomials (forms) of degree $d$ stabilizes. We study the limits of the cones of symmetric nonnegative polynomials and symmetric sums of squares, when expressed in power-mean or monomial-mean basis. These limits correspond to forms with stable expression in power-mean (or monomial-mean) polynomials that are globally nonnegative (resp. sums of squares) regardless of the number of variables. We introduce partial symmetry reduction to describe the limit cone of symmetric sums of squares, and reprove a result of arXiv:1205.3102v4 that limits of symmetric nonnegative polynomials and sums of squares agree in degree $4$. We use tropicalization of the dual cones, which was first in the context of comparing nonnegative polynomials and sums of squares in arXiv:2203.06291, to show differences between cones of symmetric polynomials and sums of squares starting in degree 6, which disproves a conjecture of arXiv:1205.3102v4. For even symmetric nonnegative forms and sums of squares we show that the cones agree for degree at most 8, and are different starting with degree 10. We also find, via tropicalization, explicit examples of symmetric forms that are nonnegative but not sums of squares in the limit.

Summary

We haven't generated a summary for this paper yet.