We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in $\ZZ[x]$; we include a new bound which was latent in a paper by Mignotte, and a few improvements to some existing bounds. We compare these bounds, and for each bound give explicit examples where that bound is the best; thus showing that no one bound 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.
Bounds on Factors in Z[x]
ABBOTT, JOHN ANTHONY
2013-01-01
Abstract
We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in $\ZZ[x]$; we include a new bound which was latent in a paper by Mignotte, and a few improvements to some existing bounds. We compare these bounds, and for each bound give explicit examples where that bound is the best; thus showing that no one bound 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.
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