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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


The Dedekind-Mertens lemma and the contents of polynomials

Author: David E. Rush
Journal: Proc. Amer. Math. Soc. 128 (2000), 2879-2884
MSC (1991): Primary 13A15, 13B25, 13B02
Published electronically: April 7, 2000
MathSciNet review: 1670427
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Let $R$ be a commutative ring, let $X$be an indeterminate, and let $g \in R[X]$. There has been much recent work concerned with determining the Dedekind-Mertens number $\mu_R(g)$=min $\{ k \in \mathbb{N}\; \vert \; c_R(f)^{k-1} c_R(fg) = c_R(f)^{k} c_R(g) \mbox{ for all } f \in R[X] \}$, especially on determining when $\mu_R(g)$ = $1$. In this note we introduce a universal Dedekind-Mertens number $u \mu_R(g)$, which takes into account the fact that $\mu_S(g)$ $\leq$ deg($g$) + $1$ for any ring $S$ containing $R$ as a subring, and show that $u \mu_R(g)$behaves more predictably than $\mu_R(g)$.

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Additional Information

David E. Rush
Affiliation: Department of Mathematics, University of California, Riverside, California 92507

PII: S 0002-9939(00)05394-6
Keywords: Dedekind-Mertens lemma, Dedekind-Mertens number, content of a polynomial
Received by editor(s): September 16, 1998
Received by editor(s) in revised form: November 29, 1998
Published electronically: April 7, 2000
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society

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