Related papers: A remark on sums of squares of complex vector fiel…

The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the…

We give a bijective parameter representation for a sum of squares of numbers being equal to another sum of squares of numbers.

Kato's second representation theorem is generalized to solvable sesquilinear forms. These forms need not be non-negative nor symmetric. The representation considered holds for a subclass of solvable forms (called hyper-solvable), precisely…

The paper studies the complex 1-dimensional polynomial vector fields with real coefficients under topological orbital equivalence preserving the separatrices of the pole at infinity. The number of generic strata is determined, and a…

The quotient class of a non-archimedean field is the set of cosets with respect to all of its additive convex subgroups. The algebraic operations on the quotient class are the Minkowski sum and product. We study the algebraic laws of these…

Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their…

We introduce a more general discrete fractional operator, given by convex linear combination of the delta and nabla fractional sums. Fundamental properties of the new fractional operator are proved. As particular cases, results on delta and…

The problem of characterizing a real polynomial $f$ as a sum of squares of polynomials on a real algebraic variety $V$ dates back to the pioneering work of Hilbert in [Mathematische Annalen 32.3 (1888): 342-350]. In this paper, we…

Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.

This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.

David Hilbert proved that a non-negative real quartic form f(x,y,z) is the sum of three squares of quadratic forms. We give a new proof which shows that if the complex plane curve Q defined by f is smooth, then f has exactly 8 such…

We prove a result about the non-existence of certain sums-of-squares formulas over a field. This generalizes an old theorem which used topological K-theory to obtain obstruction conditions when the field is the real numbers. Our result…

To prove that a polynomial is nonnegative on R^n one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than…

Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X is a sum of squares of linear forms if and only if X is a variety of…

We investigate linear operators $A:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$. We give explicit operators $A$ such that, for fixed $d\in\mathbb{N}_0$ and closed $K\subseteq\mathbb{R}^n$, $e^A\mathrm{Pos}(K)_{\leq…

A Cauchy type integral operator is associated to a class of integrable vector fields with complex coefficients. Properties of the integral operator are used to deduce Holder solvability of semilinear equations and a strong similarity…

A partial field is an algebraic object that allows one to simultaneously abstract several different representability properties of matroids. In this paper we study partial fields as algebraic objects in their own right. We characterize the…

We find the minimum scale factor, for which the nonnegative B\"ottcher-Wenzel biquadratic form becomes a sum of squares (sos). To this we give the primal and dual solutions for the underlying semide finite program. Moreover, for special…

Given a complex, elliptic coefficient function we investigate for which values of $p$ the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly…

We provide explicit conditions for a real polynomial $f$ of degree 2d to be a sum of squares (s.o.s.), stated only in terms of the coefficients of $f$, i.e. with no lifting. All conditions are simple and provide an explicit description of a…