Interpolation over ZZ and torsion in class groups

TL;DR

This paper presents an elementary proof of an interpolation theorem for homogeneous polynomials over integers and PIDs, avoiding reliance on the torsion property of class groups, with implications for algebraic number theory.

Contribution

Provides a new, elementary proof of an interpolation result that traditionally depends on the torsion nature of class groups.

Findings

Elementary proof of the interpolation theorem

Applicable to PIDs with finite residue fields

Avoids reliance on class group torsion property

Abstract

We prove an interpolation result for homogeneous polynomials over the integers, or more generally for PIDs with finite residue fields. Previous proofs of this result use the well-known but nontrivial fact that class groups of rings of integers are torsion. In this short note, we provide an independent proof using elementary techniques.