Non-absolutely irreducible elements in the ring of integer-valued polynomials

Irreducible Polynomials and Factorization Properties of the Ring of Integer-Valued Polynomials

On the Ring of Integer-valued Quasi-polynomials

The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized GCD are presented. Applications to finite simple continued fraction expansion and Smith normal form of integral matrices with integer parameters are also given.

Integer-valued Polynomials

Let R be a Krull ring with quotient field K and a1, . . . , an in R. If and only if the ai are pairwise incongruent mod every height 1 prime ideal of infinite index in R does there exist for all values b1, . . . , bn in R an interpolating integer-valued polynomial, i.e., an f ∈ K[x] with f(ai) = bi and f(R) ⊆ R. If S is an infinite subring of a discrete valuation ring Rv with quotient field K a...

Integer-valued Polynomials on Algebras

Let D be a domain with quotient field K and A a D-algebra. A polynomial with coefficients in K that maps every element of A to an element of A is called integer-valued on A. For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitrary ideal of D we show I -adic continuity of integer-valued polynomials on A. For Noetherian one-d...

Root separation for irreducible integer polynomials

where the latter limsup is taken over the irreducible integer polynomials P (x) of degree d. A classical result of Mahler [10] asserts that e(d) ≤ d− 1 for all d, and it is easy to check that eirr(2) = e(2) = 1. There is only one other value of d for which e(d) or eirr(d) is known, namely d = 3, and we have eirr(3) = e(3) = 2, as proved, independently, by Evertse [9] and Schönhage [11]. For lar...