The representation of integers as sums of squares

2002 ◽  
Vol 124 (5) ◽  
pp. 1059-1081 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Sums Of Squares ◽  
Representation Of Integers

Some Generalizations of Ramanujan's Sum

1980 ◽  
Vol 32 (5) ◽  
pp. 1250-1260 ◽  
Author(s):  
K. G. Ramanathan ◽  
M. V. Subbarao
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Sums Of Squares ◽  
Modular Functions ◽  
Algebraic Number Fields ◽  
Representation Of Integers ◽  
Ramanujan’S Sum ◽  
Image Position ◽  
Ramanujan's Sum

Ramanujan's well known trigonometrical sum C(m, n) denned bywhere x runs through a reduced residue system (mod n), had been shown to occur in analytic problems concerning modular functions of one variable, by Poincaré [4]. Ramanujan, independently later, used these trigonometrical sums in his remarkable work on representation of integers as sums of squares [6]. There are various generalizations of C(m, n) in the literature (some also to algebraic number fields); see, for example, [9] which gives references to some of these. Perhaps the earliest generalization to algebraic number fields is due to H. Rademacher [5]. We here consider a novel generalization involving matrices.

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

2021 ◽  
Vol 107 ◽  
pp. 67-105
Author(s):  
Elisabeth Gaar ◽  
Daniel Krenn ◽  
Susan Margulies ◽  
Angelika Wiegele
Keyword(s):  
Semidefinite Programming ◽  
Sums Of Squares

scholarly journals Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

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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