On sums of squares in local rings

AbstractA classical problem in real geometry concerns the representation of positive semidefinite elements of a ring A as sums of squares of elements of A. If A is an excellent ring of dimension $$\ge 3$$ ≥ 3 , it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in A. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the 2-dimensional case and determine (under some mild conditions) which local excellent henselian rings A of embedding dimension 3 have the property that every positive semidefinite element of A is a sum of squares of elements of A.

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Weighted sums of squares in local rings and their completions, I

Mathematische Zeitschrift ◽

10.1007/s00209-009-0551-6 ◽

2009 ◽

Vol 266 (1) ◽

pp. 1-19 ◽

Cited By ~ 2

Author(s):

Claus Scheiderer

Keyword(s):

Sums Of Squares ◽

Weighted Sums ◽

Local Rings

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Sums of Squares and Triangular Numbers

Online Journal of Analytic Combinatorics ◽

10.15852/ojac001.a.01 ◽

2006 ◽

Cited By ~ 2

Author(s):

Hershel Farkas

Keyword(s):

Sums Of Squares ◽

Triangular Numbers

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Towards a computational proof of Vizing's conjecture using semidefinite programming and sums-of-squares

Journal of Symbolic Computation ◽

10.1016/j.jsc.2021.01.003 ◽

2021 ◽

Vol 107 ◽

pp. 67-105

Author(s):

Elisabeth Gaar ◽

Daniel Krenn ◽

Susan Margulies ◽

Angelika Wiegele

Keyword(s):

Semidefinite Programming ◽

Sums Of Squares

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Correction to: Classification of non-local rings with genus two zero-divisor graphs

Soft Computing ◽

10.1007/s00500-020-05533-z ◽

2021 ◽

Vol 25 (4) ◽

pp. 3355-3356

Author(s):

T. Asir ◽

K. Mano ◽

T. Tamizh Chelvam

Keyword(s):

Zero Divisor ◽

Local Rings ◽

Non Local ◽

Genus Two

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Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

Foundations of Computational Mathematics ◽

10.1007/s10208-021-09496-x ◽

2021 ◽

Author(s):

Mareike Dressler ◽

Adam Kurpisz ◽

Timo de Wolff

Keyword(s):

Computer Science ◽

Affine Transformation ◽

Optimization Problems ◽

Polynomial Optimization ◽

Theoretical Computer Science ◽

Sums Of Squares ◽

Theoretical Computer ◽

Complexity Bounds ◽

Polynomial Optimization Problems ◽

Transformation Of Variables

AbstractVarious key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + d . Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

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