Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-00-05394-6
Published electronically: April 7, 2000
MathSciNet review: 1670427
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

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 $\mu_R(g)$.


References [Enhancements On Off] (What's this?)

  • 1. J. T. Arnold and R. Gilmer, On the contents of polynomials, Proc. Amer. Math. Soc. 24 (1970), 556-562. MR 40:5581
  • 2. D. D. Anderson and B. J. Kang, Content formulas for polynomials and power series and complete integral closure, J. Algebra, 181 (1987), 82-94. MR 97c:13014
  • 3. W. Bruns and A. Guerrieri, The Dedekind-Mertens formula and determinantal rings, Proc. Amer. Math. Soc. 127 (1999), 657-663. MR 99f:13013
  • 4. A. Corso, W. Heinzer and C. Huneke, A generalized Dedekind-Mertens lemma and its converse, Trans. Amer. Math. Soc. 350 (1998), 5095-5106. MR 99b:13012
  • 5. A. Corso, W. Vasconcelos and R. Villarreal, Generic Gaussian ideals, J. Pure and Applied Algebra 125 (1998), 117-127. MR 98m:13014
  • 6. H. Edwards, Divisor Theory, Birkhäuser, Boston, 1990. MR 93h:11115
  • 7. R. Gilmer, A. Grams and T. Parker, Zero divisors in power series rings, J. Reine Angew. Math., 278/79 (1975), 145-164. MR 52:8117
  • 8. S. Glaz and W. Vasconcelos, The content of Gaussian polynomials, J. Algebra 202 (1998), 1-9. MR 99c:13003
  • 9. W. Heinzer and C. Huneke, Gaussian polynomials and content ideals, Proc. Amer. Math. Soc. 125 (1997), 739-745. MR 97e:13015
  • 10. W. Heinzer and C. Huneke, The Dedekind-Mertens lemma and the contents of polynomials, Proc. Amer. Math. Soc. 126 (1998), 1305-1309. MR 98j:13003
  • 11. A. Hurwitzs, Ueber einen Fundamentalsatz arithmetischen Theorie der algebaischen Gröen, Nachr. kön Ges. Wiss. Göingen, 1895, 230-240. Werke, vol. 2, 198-207.)
  • 12. F. Mertens, Über einen algebaischen satz, S.-B. Akad. Wiss. Wein Abtheilung IIa101 (1892), 1560-1566.
  • 13. D. G. Northcott, A generalization of a theorem on the content of polynomials, Proc. Camb. Philos. Soc. 55 (1959), 282-288. MR 22:1600
  • 14. H. Tsang, Gauss's Lemma, Ph.D. Thesis, University of Chicago, 1965.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13A15, 13B25, 13B02

Retrieve articles in all journals with MSC (1991): 13A15, 13B25, 13B02


Additional Information

David E. Rush
Affiliation: Department of Mathematics, University of California, Riverside, California 92507
Email: rush@math.ucr.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05394-6
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

American Mathematical Society