论文标题
双变量多项式的词素groebner基础一个单变量
Lexicographic Groebner bases of bivariate polynomials modulo a univariate one
论文作者
论文摘要
令k [x]中的t（x）为一元非恒定多项式，然后写入r = k [x] /（t）商环。考虑两个在r [y]中的两个双变量多项式A（x，y），b（x，y）。在第一部分中，t = p^e被认为是不可约多项式p的功能。引入了一种新的算法，该算法得以计算理想的最小词典基础基础（a，b，p^e）。第二部分通过通过“动态评估”的概括而实现的“局部/全局”原理进行一般的“局部/全局”原理扩展了该算法，该原理限制在迄今为止无平方的多项式t。该算法根据案例区别“可逆/尼尔氏剂”产生分裂，在经典动态评估中扩展了通常的“可逆/零”。该算法属于Euclidean家族，核心是A和B模型T的subsultant序列。尤其是不需要分解或Groebner基依据计算。理论背景依赖于Lazard的结构定理，用于两个变量中的词典GROEBNER基础。岩浆实现了实现。与Groebner Bases方法相比，基准测试明显显示出这种方法的好处，有时很重要。
Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Groebner basis of the ideal ( a, b, p^e), is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynamic evaluation", restricted so far to a polynomial T that is squarefree. The algorithm produces splittings according to the case distinction "invertible/nilpotent", extending the usual "invertible/zero" in classic dynamic evaluation. This algorithm belongs to the Euclidean family, the core being a subresultant sequence of a and b modulo T. In particular no factorization or Groebner basis computations are necessary. The theoretical background relies on Lazard's structural theorem for lexicographic Groebner bases in two variables. An implementation is realized in Magma. Benchmarks show clearly the benefit, sometimes important, of this approach compared to the Groebner bases approach.
