Bounds on Factors in Z[x]

TL;DR

This paper reviews various bounds on the sizes of coefficients in polynomial factors over integers, introduces a new bound, and provides examples illustrating why these bounds are necessary and sometimes conservative.

Contribution

It introduces a new coefficient bound in Z[x] factorization and compares it with existing bounds, including new examples of large coefficients.

Findings

No single bound is universally best for all polynomials.

Examples show factors can have unexpectedly large coefficients.

The new bound improves understanding of coefficient size limitations.

Abstract

We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in Z[x]; we include a new bound which was latent in a paper by Mignotte, and a few minor improvements to some existing bounds. We compare these bounds and show that none is universally better than the others. In the second part of the paper we give several concrete examples of factorizations where the factors have "unexpectedly" large coefficients. These examples help us understand why the bounds must be larger than you might expect, and greatly extend the collection published by Collins.