manuscripta mathematica

, Volume 98, Issue 1, pp 115–132

On regular sequences of binomials

  • Günter Scheja
  • Ortwin Scheja
  • Uwe Storch

DOI: 10.1007/s002290050129

Cite this article as:
Scheja, G., Scheja, O. & Storch, U. manuscripta math. (1999) 98: 115. doi:10.1007/s002290050129


Let F1,…,Fr denote generic polynomials, i.e. polynomials with indeterminate coefficients. In our treatment of such polynomials and their specializations at the Trieste Workshop in 1992 we determined under which conditions F1,…,Fr generate a prime ideal. This is done algebraically in terms of regular sequences and combinatorially in terms of the involved monomials. If these conditions are fulfilled, the sequence F1,…,Fr is now said to be strictly admissible.

In this paper we attend to sequences \(\) of generic binomials in T0,…,Tn with \(\). We show in Section 2 that strict admissiblility for binomials can also be characterized via {\it distinguished} sequences of exponents, a combinatorial concept due to Ch.~Delorme. The main tools of the proofs are developed in Section 3 using methods of graph theory.

In Section 1 connections to monoid theory are outlined: The generic binomials\linebreak F1,…,Fr specialize to the binomials \(\), which define the monoid algebra \(\) , K any field, of a quotient monoid W of \(\). The strict admissibility of F1,…,Fr is closely related to the property of the monoid W to be cancellative and a complete intersection. This generalizes the description of complete intersection monoids \(\) given by Ch. Delorme in 1976.

In Section 4 we discuss computational aspects and show in particular that strict admissibility for binomial sequences is decidable in polynomial time.

Mathematics Subject Classification (1991):05C38, 13C40, 13P99, 20M14 

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Günter Scheja
    • 1
  • Ortwin Scheja
    • 2
  • Uwe Storch
    • 3
  1. 1.Mathematische Fakultät der Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany. e-mail: guenter.scheja@uni.tuebingen.deDE
  2. 2.Fachbereich Mathematik der Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany. e-mail: scheja@math.uni-sb.deDE
  3. 3.Fakultät für Mathematik der Ruhr-Universität, Universitätsstraße 150, D-44801 Bochum, Germany. e-mail: uwe.storch@ruhr-uni-bochum.deDE