Sparsity in sums of squares of polynomials

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming (SDP) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy.A considerable part of this work was conducted while this author was visiting Tokyo Institute of Technology. Research supported by Kosef R004-000-2001-00200Mathematics Subject Classification (1991): 90C22, 90C26, 90C30     

Sparse linear problems of the least squares method

One gives a survey of the direct method of solving large sparse diverse overdetermined linear systems of full column rank in the least squares sense. The survey covers practically all investigations on this topic, published in the seventies and in the beginning of the eighties.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 23, pp. 219–285, 1985.     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

Sparse anti-magic squares and vertex-magic labelings of bipartite graphs

A sparse anti-magic square is an n×n array whose non-zero entries are the consecutive integers 1,…,m for some m?n² and whose row-sums and column-sums form a set of consecutive integers. We derive some basic properties of these arrays and provide constructions for several infinite families of them. Our main interest in these arrays is their application to constructing vertex-magic labelings for bipartite graphs.     

Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials

This paper presents an algorithm and its implementation in the software package NCSOStools for finding sums of Hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allows us to use semidefinite programming. Throughout the paper several examples are given illustrating the results.     

Uniform approximation on the sphere by least squares polynomials

Numerical Algorithms - The paper concerns the uniform polynomial approximation of a function f, continuous on the unit Euclidean sphere of ?3 and known only at a finite number of points that...     

Sparse univariate polynomials with many roots over finite fields

Suppose q is a prime power and $f\in {\mathbb{F}}_q[x]$ is a univariate polynomial with exactly t monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound of $2{(q-1)}^{\frac{t-2}{t-1}}$ on the number of cosets in ${\mathbb{F}}_q^{⁎}$ needed to cover the roots of f in ${\mathbb{F}}_q^{⁎}$. Here, we give explicit f with root structure approaching this bound: When q is a perfect $(t-1)$-st power we give an explicit t-nomial vanishing on $q^{\frac{t-2}{t-1}}$ distinct cosets of ${\mathbb{F}}_q^{⁎}$. Over prime fields ${\mathbb{F}}_p$, computational data we provide suggests that it is harder to construct explicit sparse polynomials with many roots. Nevertheless, assuming the Generalized Riemann Hypothesis, we find explicit trinomials having $\mathrm{\Omega }\left(\frac{\log p}{\log \log p}\right)$ distinct roots in ${\mathbb{F}}_p$.     

Semidefinite programming and sums of hermitian squares of noncommutative polynomials

An algorithm for finding sums of hermitian squares decompositions for polynomials in noncommuting variables is presented. The algorithm is based on the “Newton chip method”, a noncommutative analog of the classical Newton polytope method, and semidefinite programming.     

There are significantly more nonegative polynomials than sums of squares

We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of even powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinity. When the degree is larger than two, it follows that there are significantly more nonnegative polynomials than sums of squares and there are significantly more sums of squares than sums of even powers of linear forms. Moreover, we quantify the exact discrepancy between the cones; from our bounds it follows that the discrepancy grows as the number of variables increases.     

A note on the maximal coefficients of squares of Newman polynomials

In a recent paper [G. Yu, An upper bound for B₂[g] sets, J. Number Theory 122 (1) (2007) 211-220] Gang Yu stated the following conjecture: Let be an arbitrary sequence of polynomials with increasing degrees and all coefficients in {0,1}. If we denote by (#pi) the number of non-zero coefficients of pi, and let be the maximal coefficient of , then

(∗)     

A composition formula for squares of Hermite polynomials and its generalizations

We prove a general formula which, with appropriately chosen parameters, gives a composition formula for squares of Gould–Hopper polynomials g²_n(x,h), and hence also for Hermite polynomials. Our main tool is the classical Mehler formula, but with imaginary arguments. To cite this article: P. Graczyk, A. Nowak, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

Lovász and Schrijver (SIAM J. Optim. 1:166–190, 1991) have constructed semidefinite relaxations for the stable set polytope of a graph G = (V,E) by a sequence of lift-and-project operations; their procedure finds the stable set polytope in at most α(G) steps, where α(G) is the stability number of G. Two other hierarchies of semidefinite bounds for the stability number have been proposed by Lasserre (SIAM J. Optim. 11:796–817, 2001; Lecture Notes in Computer Science, Springer, Berlin Heidelberg New York, pp 293–303, 2001) and by de Klerk and Pasechnik (SIAM J. Optim. 12:875–892), which are based on relaxing nonnegativity of a polynomial by requiring the existence of a sum of squares decomposition. The hierarchy of Lasserre is known to converge in α(G) steps as it refines the hierarchy of Lovász and Schrijver, and de Klerk and Pasechnik conjecture that their hierarchy also finds the stability number after α(G) steps. We prove this conjecture for graphs with stability number at most 8 and we show that the hierarchy of Lasserre refines the hierarchy of de Klerk and Pasechnik.      

On the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares

A new proof of a recent theorem of Ellis, Gohberg, and Lay, which identifies the number of roots of a “continuous” matrix orthogonal polynomial in the open upper halfplane with the number of negative eigenvalues of a related integral operator is presented. A related extension problem is then formulated and solved in assorted classes of functions which are analytic in the open upper half plane, apart from a finite number of poles. A discrete analogue of this extension problem is also formulated and solved. © 1994 John Wiley & Sons, Inc.     

Sparse polynomials, redundant bases, gauss periods, and efficient exponentiation of primitive elements for small characteristic finite fields

Gauss periods give an exponentiation algorithm that is fast for many finite fields but slow for many other fields. The current paper presents a different method for construction of elements that yield a fast exponentiation algorithm for finite fields where the Gauss period method is slow or does not work. The basic idea is to use elements of low multiplicative order and search for primitive elements that are binomial or trinomial of these elements. Computational experiments indicate that such primitive elements exist, and it is shown that they can be exponentiated fast.