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The Dedekind-Mertens lemma
and the contents of polynomials


Authors: William Heinzer and Craig Huneke
Journal: Proc. Amer. Math. Soc. 126 (1998), 1305-1309
MSC (1991): Primary 13A15, 13B25, 13G05, 13H10
DOI: https://doi.org/10.1090/S0002-9939-98-04165-3
MathSciNet review: 1425124
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a sharpening of the Dedekind-Mertens Lemma relating the contents of two polynomials to the content of their product. We show that for a polynomial $g$ the integer $1 + \deg (g)$ in the Dedekind-Mertens Lemma may be replaced by the number of local generators of the content of $g$. We also raise a question concerning the converse.


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

  • [AG] J. Arnold and R. Gilmer, On the contents of polynomials, Proc. Amer. Math. Soc. 24 (1970), 556-562. MR 40:5581
  • [AK] D. D. Anderson and B. G. Kang, Content formulas for polynomials and power series and complete integral closure, J. Algebra 181 (1996), 82-94. MR 97c:13014
  • [CVV] A. Corso, W. Vasconcelos and R. Villarreal, Generic Gaussian ideals, J. Pure Appl. Algebra (to appear).
  • [De] R. Dedekind, Über einen arithmetischen Satz von Gauss, Gesammelte Werke XXII, Vol 2, Mitt. Deutsch. Math. Ges. Prague (1892), 1-11.
  • [Ed] H. Edwards, Divisor Theory, Birkhäuser, Boston, 1990. MR 93h:11115
  • [GGP] R. Gilmer, A. Grams, and T. Parker, Zero divisors in power series rings, Jour. reine angew. Math. 278/79 (1975), 145-164. MR 52:8117
  • [GV] S. Glaz and W. Vasconcelos, The content of Gaussian polynomials, J. Algebra (to appear).
  • [HH] W. Heinzer and C. Huneke, Gaussian polynomials and content ideals, Proc. Amer. Math. Soc. 125 (1997), 739-745. MR 97e:13015
  • [Hu] A. Hurwitz, Ueber einen Fundamentalsatz der arithmetischen Theorie der algebraischen Grössen, Nachr. kön Ges. Wiss. Göttingen (1895), 230-240.
  • [Kro] L. Kronecker, Zur Theorie der Formen höherer Stufen, Monatsber Akad. Wiss. Berlin (1883), 957-960.
  • [Kru] W. Krull, Idealtheorie, Zweite, ergänzte Auflage, Springer-Verlag, Berlin, 1968. MR 37:5197
  • [M] H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986. MR 88h:13001
  • [Mer] F. Mertens, Über einen algebraischen Satz, S.-B. Akad. Wiss. Wien (2a) 101 (1892), 1560-1566.
  • [N] D.G. Northcott, A generalization of a theorem on the content of polynomials, Proc. Cambridge Philos. Soc. 55 (1959), 282-288. MR 22:1600
  • [Pr] H. Prüfer, Untersuchungen über Teilbarkeitseigenschaften in Körpern, J. Reine Angew. Math. 168 (1932), 1-36.

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Additional Information

William Heinzer
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395
Email: heinzer@math.purdue.edu

Craig Huneke
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395
Email: huneke@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04165-3
Keywords: Dedekind-Mertens Lemma, content of a polynomial
Received by editor(s): July 9, 1996
Received by editor(s) in revised form: October 23, 1996
Additional Notes: The second author was partially supported by the NSF
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1998 American Mathematical Society

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