Development of high performance computer algebra software

我们开发了以下新算法并在RISA/ASIR中实现：(1)小有限域上的多项式时间二元多项式因式分解算法，(2)小有限域上多项式理想的极小素因数的计算算法，(3)全局b-函数的计算算法，(4)多元超几何函数的有效数值计算算法，(5)广义超几何函数的二次关系的推导方法，(6)幂级数环上微分算子环上的切锥算法。我们在RISA/ASIR中实现了以下新工具：(1)用新的数据表示来计算Groebner基的新包；(2)代数数域中有效计算Groebner基的新包。我们给出了以下应用：(1)精确地解决了与量子计算有关的多项式系统。(2)证明了中心电荷为6/5的W_3代数是格点算子代数的子代数，并对它的不可约模进行了分类。

We developed the following new algorithms and implemented them in Risa/Asir :(1)a polynomial-time bivariate polynomial factorization algorithm over small finite fields,(2)an algorithm for computing minimal prime divisors of a polynomial ideal over small finite fields,(3)an algorithm for computing the global b-function,(4)an efficient algorithm for numerical computation of multivariate hypergeometric functions,(5)a method for deriving quadratic relations for generalized hypergeometric functions,(6)a tangent cone algorithm in the ring of differential operators over power series ring.We implemented the following new facilities in Risa/Asir :(1)a new package for computing Groebner basis with a new data representation,(2)a new package for efficient computation in algebraic number fields.We gave the following applications.(1)We exactly solved a polynomial system concerned with quantum computing.(2)We proved that W_3 algebra of central charge 6/5 is realized as a subalgebra of a lattice vertex operator algebra and classify its irreducible modules.