We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with posit...

Generalized nonnegative polynomials are defined as products of nonnegative polynomials raised to positive real powers. The generalized degree can be defined in a natural way. In this paper we extend quadrature sums involving pth powers of polynomials to those for generalized polynomials.

1. The problem. In [l], the following problem is posed: Let 0<A<1. Find the smallest integer re for which there exist re real numbers ai, a2, ■ ■ • , an such that the polynomial (y2 — 2Ay + T) ' II?-1 Cy+0») has all its coefficients non-negative. If N is such an integer, does there exist some real number a such that (y2 — 2Ay + l) •(y+a)N has only non-negative coefficients.1 In the process of g...

This overview is intended to provide an “ atlas ” of what is known about approximations of the cone of positive polynomials (on a semialgebraic set KS) by various preorderings (or the corresponding module versions). These approximations depend on the description S of KS, the dimension of the semi-algebraic set KS, intrinsic geometric properties of KS (e.g. compact or unbounded), and special pro...

In 1888, Hilbert described how to find real polynomials which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert’s construction and present many such polynomials. 1. History and Overview A real polynomial f(x1, . . . , xn) is psd or positive if f...

In 1942 I.J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a positive linear combination of the Gegenbauer polynomials. In this paper we extend Schoenberg’s theorem for multivariate Gegenbauer polynomials. This extension derives new positive semidefinite constraints for the distance distribution which can be applied for spherical codes.

where S is an n × n positive definite Hermitian matrix, dS is equivalent to the dH in (1) restricted in the subset of positive definite Hermitian matrices, and E+ is a subset of R+. The integral H(E) is in some sense the limiting case of L(E+), since H(E) can be evaluated by Hermitian polynomials, while L(E+) by Laguerre polynomials in the same way [8]. It is a classical result that Hermitian p...

We discuss the role of additive polynomials and p-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such rings. We prove the existence of Frobenius-closed bases of algebraic function fields F |K in one ...