Groebner basis of toric ideals on lattices and its applications to mathematical physics

Groebner基在各个领域有着广泛的应用。我们的研究出发点是计算列联表集合的马尔可夫基，列联表的元素为非负整数，列和为固定值。由于马尔可夫基与Groebner基密切相关，我们只需计算由这些表定义的整数上多项式环的理想的Groebner基。作为这一研究路线的结果，我们用复曲面理想算法计算了三向列联表的3 × 3 × 4、3 × 4 × 4、4 × 4 × 4等Groebner基。证明了由列联表导出的梯型和格基的一些引理。然后，我们对这些列联表的集合进行随机游走，得到了给定真实的数据的p值等结果。在量子计算机研究方面，我们编制了一个用经典计算机模拟量子计算机的程序，并用它模拟了一个求序问题的算法。并利用整数规划方法求出了该问题的最优解.这使我们在一般情况下的最优解的一些预测。此外，通过定义量子算法的酉矩阵，我们研究了列联表集上的量子随机游动。

There are many applications of Groebner basis to various fields. The starting point of our research is to calculate Markov basis for the set of contingency tables, which have non-negative integers as their elements and are fixedwith some given row and column sums. Since Markov basis is closely related to Groebner basis, we need only to calculate Groebner basis for the ideal of a polynomial ring over integer defined by these tables. As our results of this research line, we calculated the Groebner basis of 3x3x4, 3x4x4, 4x4x4, etc for three way contingency tables byuse of an algorithmwith toric ideals. Also, we proved some lemma of echelon form and lattice basis coming from contingency tables. And then, we performed random walks on the set of these contingency tables, and obtained results such as p-value for given real data. This is sufficient to solve the Sturmfels conjecture.As for our quantum computer research, we made a program of emulating quantum computers by aclassical computer, which were used to simulate an algorithm of the order-finding problem. And, we calculated the optimal solutions for this problem by use of the integer programmingmethod!It leads us to some prediction about the optimal solutions in general case. Moreover, we did aresearch of quantum randomwalks on the set of contingency tables by defining unitary matricesfor this quantum algorithm.