Le Nullstellensatz de Hilbert et les Polynomes à Valeurs Entières.
Ideaux de polynomes et ideaux de valeurs.
Polynomes à valeurs entières et propriété de Skolem.
Les idéaux premiers de l'anneau des polynomes à valeurs entières.
Dérivées et différences divisées à valeurs entières
An overview of some recent developments on integer-valued polynomials: Answers and Questions
The purpose of my talk is to give an overview of some more or less recent developments on integer-valued polynomials and, doing so, to emphasize that integer-valued polynomials really occur in different areas: combinatorics, arithmetic, number theory, commutative and non-commutative algebra, topology, ultrametric analysis, and dynamics. I will show that several answers were given to open problems, and I will raise also some new questions.
On the ultrametric Stone-Weierstrass theorem and Mahler's expansion
We describe an ultrametric version of the Stone-Weierstrass theorem, without any assumption on the residue field. If is a subset of a rank-one valuation domain , we show that the ring of polynomial functions is dense in the ring of continuous functions from to if and only if the topological closure of in the completion of is compact. We then show how to expand continuous functions in sums of polynomials.
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