关于矩阵的Gröbner复杂性
On the Gröbner complexity of matrices
摘要：
本文证明了：如果对一个整数矩阵A，伴随环面理想\Ideal_A的泛Gröbner基与A的Graver基一致，则它的高阶Lawrence提升的Gröbner复杂度u（A）与Graver复杂度g（A）也一致.我们得出结论，对于矩阵A_{3\times 3}和A_{3\times 4}，定义3 \times 3和3 \times 4运输问题，我们有u（A_{3\times 3}）=g（A_{3\times 3}）=9和u（A_{3\times 4}）=g（A_{3\times 4}）MQ \geq 27。进一步证明了对正整数a，b，u（A_{a,b}）=g（A_{a,b}）=2（a+b）/ \gcd（a，b），A_{a,b} =（\begin{smallmatrix} 1 & 1 & 1 & 1 0 & a & b & a+b \end{smallmatrix}）.
英文摘要：
In this paper we show that if for an integer matrix A the universal Gröbner basis of the associated toric ideal \Ideal_A coincides with the Graver basis of A, then the Gröbner 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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