On the Gröbner complexity of matrices

In this paper we show that if for an integer matrix AA the universal Gröbner basis of the associated toric ideal IAIA coincides with the Graver basis of AA, then the Gröbner complexity u(A)u(A) and the Graver complexity g(A)g(A) of its higher Lawrence liftings agree, too. In fact, if the universal Gröbner basis of IAIA coincides with the Graver basis of AA, then also the more general complexities u(A,B)u(A,B) and g(A,B)g(A,B) agree for arbitrary BB. We conclude that for the matrices A3×3A3×3 and A3×4A3×4, defining the 3×3 and 3×4 transportation problems, we have u(A3×3)=g(A3×3)=9u(A3×3)=g(A3×3)=9 and u(A3×4)=g(A3×4)≥27u(A3×4)=g(A3×4)≥27. Moreover, we prove that u(Aa,b)=g(Aa,b)=2(a+b)/gcd(a,b)u(Aa,b)=g(Aa,b)=2(a+b)/gcd(a,b) for positive integers a,ba,b and Aa,b=(11110aba+b).