关于通过Higman线性化实现高效非交换多项式因式分解
On Efficient Noncommutative Polynomial Factorization via Higman Linearization
AI总结：
本文提出一种随机多项式时间算法，利用Higman线性化将非交换代数分支程序转换为线性矩阵，进而实现非交换多项式的因式分解，输出各不可约因子。
摘要：
本文研究了在非交换变量x1,x2,...,xn上的自由非交换环F中，对多项式进行高效因式分解的问题，其中F是域F上的多项式环。我们获得如下结果：给定一个规模为s的非交换代数分支程序，它计算F中的一个非交换多项式f作为输入，其中F=Fq是有限域，我们给出一个随机算法，其运行时间为s、n和log q的多项式，该算法计算多项式f的因式分解，将其表示为乘积f=f1f2...fr，其中每个fi是不可约多项式，并以非交换代数分支程序的形式输出。该算法首先利用Higman多项式线性化，将计算f的给定代数分支程序转换为线性矩阵L。然后我们对线性矩阵L进行因式分解，并恢复多项式f的因式分解。我们使用了Cohn自由理想环理论的基本元素，结合Ronyai的随机多项式时间算法，该算法用于计算有限域上矩阵集合的不变子空间。
英文摘要：
In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F of polynomials in noncommuting variables x1,x2,...,xn over the field F. We obtain the following result Given a noncommutative algebraic branching program of size s computing a noncommutative polynomial f in F as input, where F=Fq is a finite field, we give a randomized algorithm that runs in time polynomial in s, n and log q that computes a factorization of the polynomial f as a product f=f1f2...fr, where each fi is an irreducible polynomial that is output as a noncommutative algebraic branching program. The algorithm works by first transforming the given algebraic branching program computing f into a linear matrix L using Higman's linearization of polynomials. We then factorize the linear matrix L and recover the factorization of the polynomial f. We use basic elements from Cohn's theory of free ideals rings combined with Ronyai's randomized polynomial-time algorithm for computing invariant subspaces of a collection of matrices over finite fields.
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