论文标题
通过希格曼线性化有效的非交通性多项式分解
On Efficient Noncommutative Polynomial Factorization via Higman Linearization
论文作者
论文摘要
在本文中，我们研究了在非交易变量中多项式的自由非交通环f中有效分级多项式的问题 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非共同代数分支计划。 该算法通过首先使用Higman的多项式线性化将给定代数的分支程序计算为线性矩阵L来实现。然后，我们将线性矩阵L分解并恢复多项式f的分解。我们使用Cohn的自由理想戒指理论中的基本元素，结合了罗尼伊的随机多项式时间算法来计算有限磁场上矩阵集合的不变子空间。
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.
