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A characterization of partition polynomials and good Bernoulli trial measures in many symbols

Andrew Yingst (2014)

Colloquium Mathematicae

Consider an experiment with d+1 possible outcomes, d of which occur with probabilities x , . . . , 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 , . . . , 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.

A class of irreducible polynomials

Joshua Harrington, Lenny Jones (2013)

Colloquium Mathematicae

Let f ( x ) = x + k n - 1 x n - 1 + k n - 2 x n - 2 + + k x + k [ x ] , where 3 k n - 1 k n - 2 k k 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.

A generalization of a theorem of Schinzel

Georges Rhin (2004)

Colloquium Mathematicae

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.

A new exceptional polynomial for the integer transfinite diameter of [ 0 , 1 ]

Qiang Wu (2003)

Journal de théorie des nombres de Bordeaux

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.

A note on the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

Let p(z) be a polynomial of the form p ( z ) = j = 0 n a j z j , a j - 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.

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