TY - JOUR

T1 - A note on mediated simplices

AU - Powers, Victoria

AU - Reznick, Bruce

N1 - Funding Information:
In 1989, the second author considered [14] a class of homogeneous polynomials (forms) which had arisen in the study of Hilbert's 17th Problem as monomial substitutions into the arithmetic-geometric inequality. The goal was to determine when such a form, which must be positive semidefinite, had a representation as a sum of squares of forms. The answer was a necessary and sufficient condition involving the arrangement of lattice points in the simplex whose vertices were the n-tuples of the exponents used in the substitution. Further, a claim was made in [14], and not proven, that sufficiently large dilations of any such simplex will also satisfy this condition. The aim of this short note is to prove the claim, and provide further context for the result, both in the study of Hilbert's 17th Problem and the study of lattice point simplices. The second author is happy to acknowledge that the return to this claim was triggered by two nearly simultaneous events: an invitation to speak at the 2019 SIAM Conference on Applied Algebraic Geometry, and a request from Jie Wang for a copy of [15], which was announced in [14] but never written. The second author was supported in part by Simons Collaboration Grant 280987.

PY - 2021/7

Y1 - 2021/7

N2 - Many homogeneous polynomials that arise in the study of sums of squares and Hilbert's 17th problem are those formed by monomial substitutions into the arithmetic-geometric inequality. In 1989, Reznick [14] gave a necessary and sufficient condition for such a form to have a representation as a sum of squares of forms, involving the arrangement of lattice points in the simplex whose vertices were the n-tuples of the exponents used in the substitution. Further, a claim was made, and not proven, that sufficiently large dilations of any such simplex will also satisfy this condition. The aim of this short note is to prove the claim, and provide further context for the result, both in the study of Hilbert's 17th Problem and the study of lattice point simplices.

AB - Many homogeneous polynomials that arise in the study of sums of squares and Hilbert's 17th problem are those formed by monomial substitutions into the arithmetic-geometric inequality. In 1989, Reznick [14] gave a necessary and sufficient condition for such a form to have a representation as a sum of squares of forms, involving the arrangement of lattice points in the simplex whose vertices were the n-tuples of the exponents used in the substitution. Further, a claim was made, and not proven, that sufficiently large dilations of any such simplex will also satisfy this condition. The aim of this short note is to prove the claim, and provide further context for the result, both in the study of Hilbert's 17th Problem and the study of lattice point simplices.

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U2 - 10.1016/j.jpaa.2020.106608

DO - 10.1016/j.jpaa.2020.106608

M3 - Article

AN - SCOPUS:85095597155

SN - 0022-4049

VL - 225

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 7

M1 - 106608

ER -