A new lower bound construction for commutative thue systems with applications

  • Authors:

  • Chee K. Yap

  • Affiliations:

  • Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 1991

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Abstract

For n =1, d = 2, we describe a commutative Thue system that has ~2n variables and O(n) rules, each rule of size d + O(1) and that counts to d^2^n in a certain technical sense. This gives a more ''efficient'' alternative to a well-known construction of Mayr and Meyer. Using this construction, we sharpen the known double-exponential lower bounds for the maximum degrees D(n, d), I(n, d), S(n, d) associated (respectively) with Grobner bases, ideal membership problem and the syzygy basis problem: D(n,d)=S(n,d)=d^2^^^m,I(n,d)=d^2^^^m, where m~n/2, and n, d sufficiently large. For comparison, it was known that D(n, d) @? d^2^n and I(n, d) @? (2d)^2^n.