Paul-Jean Cahen

Integer-Valued Polynomials

Coefficients and values Additive structure Stone-Weierstrass Integer-valued polynomials on a subset Prime ideals Multiplicative properties Skolem properties Invertible ideals and the Picard group Integer-valued derivatives and finite differences Integer-valued rational functions Integer-valued polynomials in several indeterminates References List of symbols Index.

Elasticity for integral-valued polynomials

The elasticity of a domain is the upper bound of the ratios of lengths of two decompositions in irreducible factors of nonzero nonunit elements. We show that for a large class of Noetherian domains, including any domain contained in a number field (but not a field), the elasticity of the ring of integral-valued polynomials is infinite.

FINITELY VALUATIVE DOMAINS

For a pair of rings S ⊆ T and a nonnegative integer n, an element t ∈ T\S is said to be within n steps of S if there is a saturated chain of rings S = S0 ⊊ S1 ⊊ ⋯ ⊊ Sm = S[t] with length m ≤ n. An integral domain R is said to be n-valuative (respectively, finitely valuative) if for each nonzero element u in its quotient field, at least one of u and u-1 is within n (respectively, finitely many) steps of R. The integral closure of a finitely valuative domain is a Prufer domain. Moreover, an n-valuative domain has at most 2n + 1 maximal ideals; and an n-valuative domain with 2n + 1 maximal ideals must be a Prufer domain.

Rings of integer-valued rational functions

Abstract Let D be an integral domain which differs from its quotient field K . The ring of integer-valued rational functions of D on a subset E of D is defined as Int R ( E , D ) = f ( X ) ∈ K ( X )| f ( E ) ⊂-. We write Int R ( D ) for Int R ( D , D ). It is easy to see that Int R ( D ) is strictly larger than the more familiar ring Int ( D ) of integer-valued polynomials precisely when there exists a polynomial f ( X ) ∈ D [ X ] such that f ( d ) is a unit in D for each d ∈ D . In fact, there arc striking differences between Int R ( D ) and Int ( D ) in many of the cases where they are not equal. Rings of integer-valued rational functions have been studied in at least two previous papers. The purpose of this note is to consolidate and greatly expand the results of these papers. Among the topics included, we give conditions so that Int R ( E , D ) is a Prufer domain, we study the value ideals of Int R ( E , D ) (for example, we show that Int R ( K , D ) satisfies the strong Skolem property provided it is a Prufer domain), and we study the prime ideals of Int R ( E , D ) (for example, we show that if V is a valuation domain, then each prime ideal of Int R ( V ) above the maximal ideal m of V is maximal if and only if m is principal).

What You Should Know About Integer-Valued Polynomials

Abstract The authors wish to celebrate the centenary of Pólyas paper Ueber ganzwertige ganze funktionen where first explicitly appeared the term “integer-valued polynomials.” This survey is focused on the emblematic example of the ring Int(ℤ) formed by the polynomials with rational coefficients taking integer values on the integers. This ring has surprising algebraic properties, often obtained by means of analytical properties. Yet, the article mentions also several extensions, either by considering integer-valued polynomials on a subset of ℤ, or by replacing ℤ by the ring of integers of a number field.

Skolem properties, value-functions, and divisorial ideals

Let D be the ring of integers of a number field K. It is well known that the ring Int(D) = {f ϵ K[X] ¦ f (D) ⊆ D} of integer-valued polynomials on D is a Prufer domain. Here we study the divisorial ideals of Int(D) and prove in particular that Int(D) has no divisorial prime ideal. We begin with the local case. We show that, if V is a rank-one discrete valuation domain with finite residue field, then the unitary ideals of Int(V) (that is, the ideals containing nonzero constants) are entirely determined by their values on the completion of V. This improves on the Skolem properties which only deal with finitely generated ideals. We then globalize and consider a Dedekind domain D with finite residue fields. We show that a prime ideal of t(D) is invertible if and only if it is divisorial, and also, in the case where the characteristic of D is 0, if and only if it is an upper to zero which is maximal.

Bhargava's Early Work: The Genesis of P-Orderings

Abstract As an undergraduate student, Manjul Bhargava gave a full answer to a question on polynomial functions on the integers. He immediately generalized this study to finite principal ideal rings, thanks to the amazingly simple notion of P-ordering. This tool, together with a beautiful generalization of factorials, allowed him to generalize many classical theorems. It turned out to be extremely useful for the study of integer-valued polynomials on subsets.

Rings, Modules, Algebras

Many problems in commutative algebra treat various ways that a fixed ideal (or each ideal of a given class of ideals) of a commutative ring can be decomposed. Generally speaking, early problems of this type that arose from algebraic geometry concerned representations of ideals as intersections, while those arising from algebraic number theory involved representations in terms of products.