A universal coefficient theorem for Gauss's Lemma

TL;DR

This paper generalizes Gauss's Lemma by providing a recursive construction of polynomials that relate the coefficients of polynomial products and their factors, with explicit degree bounds.

Contribution

It introduces a universal coefficient theorem for Gauss's Lemma, offering a recursive method to construct polynomials linking coefficients of factorizations.

Findings

Provides explicit polynomial constructions with degree bounds

Establishes a recursive framework for coefficient relations

Extends Gauss's Lemma to a more general algebraic setting

Abstract

We prove a version of Gauss's Lemma. It recursively constructs polynomials {c_k} for k=0,1,...,m+n, in Z[a_i,A_i,b_j,B_j] for i=0,...,m, and j=0,1,...,n, having degree at most (m+n choose m) in each of the four variable sets, such that whenever {A_i},{B_j},{C_k} are the coefficients of polynomials A(X),B(X),C(X) with C(X)=A(X)B(X) and 1 = a_0 A_0 +...+ a_m A_m = b_0 B_0 +...+ b_n B_n, then one also has 1 = c_0 C_0 +...+ c_{m+n} C_{m+n}.