doi:10.1016/j.jsc.2005.11.003 
Copyright © 2005 Elsevier Ltd All rights reserved.
The lex game and some applications
Bálint Felszeghy
, 
, Balázs Ráth and Lajos Rónyai
Computer and Automation Institute, Hungarian Academy of Sciences, Hungary
Department of Algebra, Budapest University of Technology and Economics, Hungary
Received 24 May 2004;
accepted 4 September 2005.
Available online 10 January 2006.
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Abstract
Let 
be a field, V a finite subset of 
. We introduce the lex game, which yields a combinatorial description of the lexicographic standard monomials of the ideal I(V) of polynomials vanishing on V.
As a consequence, we obtain a fast algorithm which computes the lexicographic standard monomials of I(V).
We apply the lex game to calculate explicitly the standard monomials for special types of subsets of {0,1}n. For 
let VD denote the vectors 
in which the number of ones (the Hamming weight of 
) is in D. We calculate the lexicographic standard monomials of VD, where
for
fixed with
0≤d<r and
0<ℓ<r. This extends the results of [Anstee, R.P., Rónyai, L., Sali, A., 2002. Shattering news. Graphs and Combinatorics 18, 59–73, Friedl, K., Hegedűs, G., Rónyai, L., Gröbner bases for complete
l-wide families (in press) and Hegedűs, G., Rónyai, L., 2003. Gröbner bases for complete uniform families. Journal of Algebraic Combinatorics 17, 171–180].
Keywords: Standard monomials; Gröbner basis; Combinatorial algorithm
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