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An overview of some recent developments on integer-valued polynomials: Answers and Questions

Jean-Luc Chabert (2010)

Actes des rencontres du CIRM

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.

An upper bound on the number of monomials in determinants of sparse matrices with symbolic entries.

Kalkbrener, M. (1997)

Mathematica Pannonica

Anneaux de Fatou

Benali Benzaghou (1968/1969)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Anneaux de 'polynômes à valeurs entières' et anneaux de Fatou

J.-L. Chabert (1971)

Bulletin de la Société Mathématique de France

Anneaux de polynômes à valeurs entières sur un anneau de valuation ou de Seidenberg

Youssef Haouat (1986)

Annales scientifiques de l'Université de Clermont. Mathématiques

Anneaux de polynômes sur un anneau de Mori

Ballet, B., Dessagnes, N. (1988)

Portugaliae mathematica

Anneaux d'invariants de groupes finis Intersections complètes

Denis Rotillon (1985)

Publications mathématiques et informatique de Rennes

Appendice à l'article précédent par Ph. Nuss: Une formule de Künneth pour la décomposition de l'homologie cyclique des algèbres commutatives.

Christian Kassel (1992)

Mathematica Scandinavica

Arithmetical quadratic surfaces of genus 0, II.

J. Brzezinski (1984)

Mathematica Scandinavica

Atomicity and the fixed divisor in certain pullback constructions

Jason Greene Boynton (2012)

Colloquium Mathematicae

Let D be an integral domain with field of fractions K. In this article, we use a certain pullback construction in the spirit of Int(E,D) that furnishes many examples of domains between D[x] and K[x] in which there are elements that do not admit a finite factorization into irreducible elements. We also define the notion of a fixed divisor for this pullback construction to characterize all of its irreducible elements and those nonzero nonunits that do admit a finite factorization into irreducibles....

Bases for integer-valued polynomials in a Galois field

Vichian Laohakosol (1998)

Acta Arithmetica

Birings and plethories of integer-valued polynomials

Jesse Elliott (2010)

Actes des rencontres du CIRM

Let $A$ and $B$ be commutative rings with identity. An $A$ - $B$ -biring is an $A$ -algebra $S$ together with a lift of the functor ${Hom}_{A} (S, -)$ from $A$ -algebras to sets to a functor from $A$ -algebras to $B$ -algebras. An $A$ -plethory is a monoid object in the monoidal category, equipped with the composition product, of $A$ - $A$ -birings. The polynomial ring $A [X]$ is an initial object in the category of such structures. The $D$ -algebra $Int (D)$ has such a structure if $D = A$ is a domain such that the natural $D$ -algebra homomorphism $θ_{n} : {⨂_{D}}_{i = 1}^{n} Int (D) \to Int (D^{n})$ is an isomorphism for...

Borel ideals in three variables.

Marinari, Maria Grazia, Ramella, Luciana (2006)

Beiträge zur Algebra und Geometrie

Bornes pour la régularité de Castelnuovo-Mumford des schémas non lisses

Amadou Lamine Fall (2009)

Annales de l’institut Fourier

Nous montrons dans cet article des bornes pour la régularité de Castelnuovo-Mumford d’un schéma admettant des singularités, en fonction des degrés des équations définissant le schéma, de sa dimension et de la dimension de son lieu singulier. Dans le cas où les singularités sont isolées, nous améliorons la borne fournie par Chardin et Ulrich et dans le cas général, nous établissons une borne doublement exponentielle en la dimension du lieu singulier.

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