The lex game and some applications

  • Authors:

  • Bálint Felszeghy;Balázs Ráth;Lajos Rónyai

  • Affiliations:

  • Computer and Automation Institute, Hungarian Academy of Sciences, Hungary and Department of Algebra, Budapest University of Technology and Economics, Hungary;Computer and Automation Institute, Hungarian Academy of Sciences, Hungary and Department of Algebra, Budapest University of Technology and Economics, Hungary;Computer and Automation Institute, Hungarian Academy of Sciences, Hungary and Department of Algebra, Budapest University of Technology and Economics, Hungary

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2006

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Abstract

Let F be a field, V a finite subset of F^n. 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 D@?Z let V"D denote the vectors y@?{0,1}^n in which the number of ones (the Hamming weight of y) is in D. We calculate the lexicographic standard monomials of V"D, where D=D(d,@?,r)={a@?Z:@?a^'@?Zwithd@?a^'@?d+@?-1anda^'=a(modr)}, for d,@?,r@?N fixed with 0@?d