Discriminants and nonnegative polynomials

Discriminants and nonnegative polynomials

For a semialgebraic set K in R, let Pd(K) = {f ∈ R[x]≤d : f(u) ≥ 0 ∀u ∈ K} be the cone of polynomials in x ∈ R of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary ∂Pd(K). When K = R n and d is even, we show that its boundary ∂Pd(K) lies on the irreducible hypersurface defined by the discriminant ∆(f) of f . When K = {x ∈ R : g1(x) = · · · = gm(x) = 0}...

Nonnegative Hall Polynomials

The number of subgroups of type n and cotype v in a finite abelian p-group of type A is a polynomial g$v(p) with integral coefficients. We prove g u v ( p ) has nonnegative coefficients for all partitions p and v if and only if no two parts of A differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certa...

Nonnegative Trigonometric Polynomials

An extremal problem for the coefficients of sine polynomials, which are nonnegativein [0, π], posed and discussed by Rogosinski and Szegő is under consideration. An analog of the Fejér-Riesz representation of nonnegativegeneral trigonometric and cosine polynomials is proved for nonnegativesine polynomials. Various extremal sine polynomials for the problem of Rogosinski and Szegő are obtained ex...

Polynomials with Nonnegative Coefficients

The authors verify the conjecture that a conjugate pair of zeros can be factored from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonnegative coefficients. The conjecture was originally posed by A. Rigler, S. Trimble, and R. Varga arising out of their work on the Beauzamy-Enflo generalization of Jensen's inequality. The conjecture was also made independ...

Discriminants of Some Painlev E Polynomials