In the classical sense, integer-valued polynomials are polynomials with rational
coefficients that map all integers to integers. Already in the 17^{th} century,
Newton used this kind of polynomials to interpolate functions. At the beginning of the
20^{th} century, Pólya, Ostrowski and Skolem discovered their role in number
theory.

In addition to their usefulness for interpolation, integer-valued polynomials have other
interesting algebraic and number theoretic properties. For example, they satisfy a $p$-adic
version of the Stone-Weierstrass property which make them suitable for approximation purposes, and
analogues of Hilbert's Nullstellensatz, the so-called Skolem properties.

We researchers of this project, Sophie Frisch, Giulio Peruginelli and Roswitha Rissner, worked in
the more general context of integer-valued polynomials on arbitrary domains. We narrowed down the
gap between sufficient and necessary conditions for Skolem properties to almost nothing, the first
significant advance on this question in decades.

Also, we were among the first to investigate polynomials which are integer-valued on matrix rings
or more general algebras. We showed that integer-valued polynomials with coefficients in a
non-commutative matrix algebra can be expressed as matrices whose entries are polynomials with
coefficients in a commutative ring. In the special case of upper triangular matrix algebras,
integer-valued polynomials coincide with polynomials whose divided differences up to order $n$ are
integer-valued. These polynomials have applications, discovered by Bhargava, to Banach spaces of
$n$-times continuously differentiable functions on a compact subset of a local field.

We also showed various other number theoretic and ring theoretic results, well-received by the
international scientific community, concerning integer-valued polynomials, their factorization,
their overrings and the PrÃ¼fer property.

- Sophie Frisch (project leader)
- Giulio Peruginelli
- Roswitha Rissner

We organized the Workshop and Conference on

** Commutative Ring Theory, Integer-valued Polynomials and Polynomial Functions**

in Graz, Austria in

**Commutative Algebra**,

- P.-J. Cahen, M. Fontana, S. Frisch, S. Glaz. Open problems in commutative ring theory , Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions", M. Fontana, S. Frisch and S. Glaz (editors), Springer 2014, pp. 293--305
- P.-J. Cahen, R. Rissner. Finiteness and Skolem closure of ideals for non-unibranched domains, Comm. Algebra, Vol. 43, No. 6 (2015), 184--191
- S. Frisch, L. Vaserstein. Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples , J. Pure Appl. Algebra, Vol. 216, No. 1 (2012), 2231--2239
- S. Frisch. Integer-valued polynomials on algebras , J. Algebra, Vol. 373 (2013), 414--425
- S. Frisch. A construction of integer-valued polynomials with prescribed sets of lengths of factorizations , Monatsh. Math., Vol. 171, No. 3-4 (2013), 341--350
- S. Frisch, D. Krenn. "Sylow $p$-groups of polynomial permutations on the integers mod $p^n$ , J. Number Theory, Vol. 133 (2013), 4188--4199
- G. Peruginelli. Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power, J. Algebra, 398 (2014), 227--242
- G. Peruginelli. Integer-valued polynomials over matrices and divided differences, Monatsh. Math., 173 (2014), no.4, 559--571
- G. Peruginelli. Integral-valued polynomials over sets of algebraic integers of bounded degree, J. Number Th., 137 (2014), 241--255
- G. Peruginelli and N. Werner. Integral closure of rings of integer-valued polynomials on algebras, Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions", M. Fontana, S. Frisch and S. Glaz (editors), Springer 2014, pp. 293--305
- G. Peruginelli. Factorization of integer-valued polynomials with square-free denominator, Comm. Algebra, 43 (2015), 197-211
- G. Peruginelli. The ring of polynomials integral-valued over a finite set of integral elements, Comm. Algebra, 8 (2016), 113-141
- R. Rissner. Null ideals of matrices over residue class rings of principal ideal domains , Linear Algebra Appl., Vol. 494 (2016), 44--69

Graz University of Technology

Department of Analysis und Number Theory (Math A)

Kopernikusgasse 24, NT02022

8010 Graz, Austria

frisch (at) blah.math.tugraz.at

Last edited: May 2016 by Roswitha Rissner