# Multihomogeneous Nonnegative Polynomials and Sums of Squares

Multihomogeneous Nonnegative Polynomials and Sums of Squares Discrete Comput Geom https://doi.org/10.1007/s00454-018-0011-3 Multihomogeneous Nonnegative Polynomials and Sums of Squares Alperen A. Ergür Received: 9 February 2016 / Revised: 10 April 2018 / Accepted: 19 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We reﬁne and extend quantitative bounds on the fraction of nonnegative polynomials that are sums of squares to the multihomogeneous case. Keywords Multihomogeneous forms · Sums of squares · Isotropic measures · Hilbert’s 17th problem Mathematics Subject Classiﬁcation Primary 52A38, Secondary 90C22 1 Introduction Let R[x]:= R[x ,..., x ] denote the ring of real n-variate polynomials and let P 1 n n,2d denote the vector space of forms (i.e. homogeneous polynomials) of degree 2d in R[x ]. A form p ∈ P is called nonnegative if p(x ) ≥ 0 for every x ∈ R .The set n,2d of nonnegative forms in P is closed under nonnegative linear combinations and n,2d thus forms a cone. We denote the cone of nonnegative degree 2d forms by Pos . n,2d A fundamental problem in polynomial optimization and real algebraic geometry is to efﬁciently certify nonnegativity for real forms, i.e., membership in Pos . n,2d Editor in Charge: János Pach Partially supported http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete & Computational Geometry Springer Journals

# Multihomogeneous Nonnegative Polynomials and Sums of Squares

, Volume OnlineFirst – Jun 5, 2018
27 pages

/lp/springer_journal/multihomogeneous-nonnegative-polynomials-and-sums-of-squares-eL0AEXG8hF
Publisher
Springer US
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Combinatorics; Computational Mathematics and Numerical Analysis
ISSN
0179-5376
eISSN
1432-0444
D.O.I.
10.1007/s00454-018-0011-3
Publisher site
See Article on Publisher Site

### Abstract

Discrete Comput Geom https://doi.org/10.1007/s00454-018-0011-3 Multihomogeneous Nonnegative Polynomials and Sums of Squares Alperen A. Ergür Received: 9 February 2016 / Revised: 10 April 2018 / Accepted: 19 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We reﬁne and extend quantitative bounds on the fraction of nonnegative polynomials that are sums of squares to the multihomogeneous case. Keywords Multihomogeneous forms · Sums of squares · Isotropic measures · Hilbert’s 17th problem Mathematics Subject Classiﬁcation Primary 52A38, Secondary 90C22 1 Introduction Let R[x]:= R[x ,..., x ] denote the ring of real n-variate polynomials and let P 1 n n,2d denote the vector space of forms (i.e. homogeneous polynomials) of degree 2d in R[x ]. A form p ∈ P is called nonnegative if p(x ) ≥ 0 for every x ∈ R .The set n,2d of nonnegative forms in P is closed under nonnegative linear combinations and n,2d thus forms a cone. We denote the cone of nonnegative degree 2d forms by Pos . n,2d A fundamental problem in polynomial optimization and real algebraic geometry is to efﬁciently certify nonnegativity for real forms, i.e., membership in Pos . n,2d Editor in Charge: János Pach Partially supported

### Journal

Discrete & Computational GeometrySpringer Journals

Published: Jun 5, 2018

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