We survey the most used bounds for positive roots of polynomials and discuss their efficiency. We obtain new inequalities on roots of polynomials. Then we give new inequalities on roots of orthogonal polynomials, obtained from the differential equations satisfied by these polynomials. Mathematics subject classification (2000): 12D10, 68W30.

In this paper we study the average L2α-norm over T -polynomials, where α is a positive integer. More precisely, we present an explicit formula for the average L2α-norm over all the polynomials of degree exactly n with coefficients in T , where T is a finite set of complex numbers, α is a positive integer, and n ≥ 0. In particular, we give a complete answer for the cases of Littlewood polynomial...

We consider bivariate real valued polynomials orthogonal with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the re...

Certificates of non-negativity are fundamental tools in optimization. A “certificate” is generally understood as an expression that makes the non-negativity of the function in question evident. Some classical certificates of non-negativity are Farkas Lemma and the S-lemma. The lift-and-project procedure can be seen as a certificate of non-negativity for affine functions over the union of two po...

In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails. Specifically, we treat a ”symmetric” polyno...

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. F...

This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given: If h(x), p(x) ∈ R[x] such that {α ∈ R | h(α) ≥ 0} = [−1, 1] and p(x) > 0 on [−1, 1], then there exist sums of squares s(x), t(x) ∈ R[x] such that p(x) = s(x) + t(x)h(x). Explicit degree bounds for s and t are given, in terms of the ...

XS f dμ. It is natural to ask if the same is true for any linear functional L : R[X ] → R which is non-negative on MS. This is the Moment Problem for the quadratic module MS. The most interesting case seems to be when S is finite. A sufficient condition for it to be true is that each f ∈ T̃S can be approximated by elements of MS in the sense that there exists an element q ∈ R[X ] such that, for ...

In this note, explicit auxiliary functions are used to get upper and lower bounds for the Mahler measure of monic irreducible totally positive polynomials with integer coefficients. These bounds involve the length and the trace of the polynomial.