### A basis for the ring of doubly integer-valued polynomials.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Consider an experiment with d+1 possible outcomes, d of which occur with probabilities $x\u2081,...,{x}_{d}$. If we consider a large number of independent occurrences of this experiment, the probability of any event in the resulting space is a polynomial in $x\u2081,...,{x}_{d}$. We characterize those polynomials which arise as the probability of such an event. We use this to characterize those x⃗ for which the measure resulting from an infinite sequence of such trials is good in the sense of Akin.

Let $f\left(x\right)=x\u207f+{k}_{n-1}{x}^{n-1}+{k}_{n-2}{x}^{n-2}+\cdots +k\u2081x+k\u2080\in \mathbb{Z}\left[x\right]$, where $3\le {k}_{n-1}\le {k}_{n-2}\le \cdots \le k\u2081\le k\u2080\le 2{k}_{n-1}-3$. We show that f(x) and f(x²) are irreducible over ℚ. Moreover, the upper bound of $2{k}_{n-1}-3$ on the coefficients of f(x) is the best possible in this situation.

We give lower bounds for the Mahler measure of totally positive algebraic integers. These bounds depend on the degree and the discriminant. Our results improve earlier ones due to A. Schinzel. The proof uses an explicit auxiliary function in two variables.

Using refinement of an algorithm given by Habsieger and Salvy to find integer polynomials with smallest sup norm on [0, 1] we extend their table of polynomials up to degree 100. For the degree 95 we find a new exceptionnal polynomial which has complex roots. Our method uses generalized Müntz-Legendre polynomials. We improve slightly the upper bound for the integer transfinite diameter of [0, 1] and give elementary proofs of lower bounds for the exponents of some critical polynomials.

Let p(z) be a polynomial of the form $p\left(z\right)={\sum}_{j=0}^{n}{a}_{j}{z}^{j}$, ${a}_{j}\in -1,1$. We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.