For n ≥1, d ≥ 2, we describe a commutative Thue system that has ∼2 n 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 Gröbner 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 ) ≤ (2 d ) 2 n .
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