Nicholas J. Werner

Integer-Valued Polynomials over Matrix Rings

When D is a commutative integral domain with field of fractions K, the ring Int(D) = {f ∈ K[x] | f(D) ⊆ D} of integer-valued polynomials over D is well-understood. This article considers the construction of integer-valued polynomials over matrix rings with entries in an integral domain. Given an integral domain D with field of fractions K, we define Int(M n (D)): = {f ∈ M n (K)[x] | f(M n (D)) ⊆ M n (D)}. We prove that Int(M n (D)) is a ring and investigate its structure and ideals. We also derive a generating set for Int(M n (ℤ)) and prove that Int(M n (ℤ)) is non-Noetherian.

Integral Closure of Rings of Integer-Valued Polynomials on Algebras

Let D be an integrally closed domain with quotient field K. Let A be a torsion-free D-algebra that is finitely generated as a D-module. For every a in A we consider its minimal polynomial μ a (X) ∈ D[X], i.e. the monic polynomial of least degree such that μ a (a) = 0. The ring Int K (A) consists of polynomials in K[X] that send elements of A back to A under evaluation. If D has finite residue rings, we show that the integral closure of Int K (A) is the ring of polynomials in K[X] which map the roots in an algebraic closure of K of all the μ a (X), a ∈ A, into elements that are integral over D. The result is obtained by identifying A with a D-subalgebra of the matrix algebra M n (K) for some n and then considering polynomials which map a matrix to a matrix integral over D. We also obtain information about polynomially dense subsets of these rings of polynomials.

POLYNOMIALS THAT KILL EACH ELEMENT OF A FINITE RING

Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings, local rings, and all finite rings of odd order). We also examine a connection to sets of integer-valued polynomials.

Split Quaternions and Integer-valued Polynomials

The integer split quaternions form a noncommutative algebra over ℤ. We describe the prime and maximal spectrum of the integer split quaternions and investigate integer-valued polynomials over this ring. We prove that the set of such polynomials forms a ring, and proceed to study its prime and maximal ideals. In particular we completely classify the primes above 0, we obtain partial characterizations of primes above odd prime integers, and we give sufficient conditions for building maximal ideals above 2.

On Least Common Multiples of Polynomials in Z/n Z[x]

Let 𝒫(n, D) be the set of all monic polynomials in ℤ/nℤ[x] of degree D. A least common multiple for 𝒫(n, D) is a monic polynomial L ∈ ℤ/nℤ[x] of minimal degree such that f divides L for all f ∈ 𝒫(n, D). A least common multiple for 𝒫(n, D) always exists, but need not be unique; however, its degree is always unique. In this article, we establish some bounds for the degree of a least common multiple for 𝒫(n, D), present constructions for common multiples in ℤ/nℤ[x], and describe a connection to rings of integer-valued polynomials over matrix rings.