On the Gröbner complexity of matrices

Details On the Gröbner complexity of matrices On the Gröbner complexity of matrices

2007-08-31

arxiv.org/abs/0708.4392v1

Mathematics

Commutative Algebra

9 pages

Scientific paper

In this paper we show that if for an integer matrix A the universal Gr\"obner basis of the associated toric ideal \Ideal_A coincides with the Graver basis of A, then the Gr\"obner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. We conclude that for the matrices A_{3\times 3} and A_{3\times 4}, defining the 3\times 3 and 3\times 4 transportation problems, we have u(A_{3\times 3})=g(A_{3\times 3})=9 and u(A_{3\times 4})=g(A_{3\times 4})\geq 27. Moreover, we prove u(A_{a,b})=g(A_{a,b})=2(a+b)/\gcd(a,b) for positive integers a,b and A_{a,b}=(\begin{smallmatrix} 1 & 1 & 1 & 1 0 & a & b & a+b \end{smallmatrix}).

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