The Dedekind-Mertens formula

and determinantal rings

Authors:
Winfried Bruns and Anna Guerrieri

Journal:
Proc. Amer. Math. Soc. **127** (1999), 657-663

MSC (1991):
Primary 13C40, 13C14, 13D40, 13P10

DOI:
https://doi.org/10.1090/S0002-9939-99-04535-9

MathSciNet review:
1468185

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a combinatorial proof of the Dedekind-Mertens formula by computing the initial ideal of the content ideal of the product of two generic polynomials. As a side effect we obtain a complete classification of the rank Cohen-Macaulay modules over the determinantal rings .

**1.**D. Bayer and M. Stillman.*Macaulay: a system for computation in algebraic geometry and commutative algebra*. Available by anonymous`ftp`from`zariski.harvard.edu`.**2.**G. Boffi, W. Bruns, and A. Guerrieri.*On the jacobian ideal of a trilinear form.*Preprint.**3.**W. Bruns and J. Herzog.*Cohen-Macaulay rings*. Cambridge University Press 1993. MR**95h:13020****4.**W. Bruns and U. Vetter.*Determinantal rings*. Lect. Notes Math.**1327**, Springer 1988. MR**89i:13001****5.**A. Conca and J. Herzog.*On the Hilbert function of determinantal rings and their canonical module.*Proc. Amer. Math. Soc.**122**(1994), 677-681. MR**95a:13016****6.**A. Corso, W. V. Vasconcelos, and R. Villareal.*Generic Gaussian Ideals*. J. Pure Appl. Algebra, to appear.**7.**D. Eisenbud.*Commutative algebra with a view towards algebraic geometry.*Springer, 1995. MR**97a:13001****8.**S. Glaz and W. V. Vasconcelos.*The content of Gaussian polynomials*. J. Algebra, to appear**9.**W. Heinzer and C. Huneke.*The Dedekind-Mertens Lemma and the contents of polynomials*. Proc. Amer. Math. Soc., to appear.**10.**W. Heinzer and C. Huneke.*Gaussian polynomials and content ideals*. Proc. Amer. Math. Soc., to appear.**11.**J. Herzog and N. V. Trung.*Gröbner bases and multiplicity of determinantal and pfaffian ideals*. Adv. in Math.**96**(1992), 1-37. MR**94a:13012**

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Additional Information

**Winfried Bruns**

Affiliation:
Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany

Email:
Winfried.Bruns@mathematik.uni-osnabrueck.de

**Anna Guerrieri**

Affiliation:
Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany

Email:
guerran@univaq.it

DOI:
https://doi.org/10.1090/S0002-9939-99-04535-9

Keywords:
Dedekind--Mertens formula,
initial ideal,
determinantal ring,
Cohen--Macaulay module

Received by editor(s):
January 22, 1997

Received by editor(s) in revised form:
June 16, 1997

Additional Notes:
The visit of the first author to the University of L’Aquila that made this paper possible was supported by the Vigoni program of the DAAD and the CRUI

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1999
American Mathematical Society