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A characterization of partition polynomials and good Bernoulli trial measures in many symbols. (English) Zbl 1303.28006
The main result of the article is the complete characterization of so called partition polynomials (in \({d>1}\) variables), i.e., such polynomials \({p(x_1,\ldots, x_d)}\) which can be expressed in the form \[ p(x_1,\ldots, x_d)=\sum_{(i_1,\ldots,i_d)\in\mathbb{Z}^d}c_{\vec{i}}^n(1-x_1-\cdots-x_d)^{n-i_1-\cdots-i_d}x_1^{i_1}\cdot\ldots\cdot x_d^{i_d} \]
for some sufficiently large \({n>0}\) and where each \({c_{\vec{i}}^n}\) is a non negative integer with \({c_{\vec{i}}^n\leq\frac{n!}{(n-i_1-\ldots-i_d)!i_1!\cdot\ldots\cdot i_d!}}.\) The proof rely on the notion of supporting polynomials and using some combinatorial calculation. In the second part of the article the author considers conditions under which the Bernoulli trial measure is good in the sense of E. Akin [Trans. Am. Math. Soc. 357, No. 7, 2681–2722 (2005; Zbl 1078.37004)].
MSC:
28A12 Contents, measures, outer measures, capacities
37A05 Dynamical aspects of measure-preserving transformations
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11C08 Polynomials in number theory
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