论文标题
表征多项式通过Galois和Galois样组的指数晶格的微不足道
Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups
论文作者
论文摘要
计算\ emph {指数晶格}的问题由单变量多项式的根部之间的所有多重关系组成，在计算机代数领域引起了很多关注。众所周知，几乎所有具有整数系数的不可约多项式仅具有微不足道的指数晶格。但是，文献中的算法很难证明通用多项式的这种微不足道。在本文中，研究了多项式的Galois组（分别分别为\ emph（分别\ emph {类似Galois样组}）和指数晶格的微不足道。 $ \ bbbq $ \ emph { - trivial}对，这是Galois组与多项式指数晶格的琐事之间关系的核心。开发了一种有效的算法来识别这些对。基于此，一种新的算法旨在证明通用不可约多项式的指数晶格的微不足道，当多项式程度变得更大时，该算法可大大改善相同类型的最先进算法。此外，引入了多项式的类似Galois类的概念。 Some properties of the Galois-like groups are proved and, more importantly, a sufficient and necessary condition is given for a polynomial (which is not necessarily irreducible) to have trivial exponent lattice.
The problem of computing \emph{the exponent lattice} which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all irreducible polynomials with integer coefficients have only trivial exponent lattices. However, the algorithms in the literature have difficulty in proving such triviality for a generic polynomial. In this paper, the relations between the Galois group (respectively, \emph{the Galois-like groups}) and the triviality of the exponent lattice of a polynomial are investigated. The $\bbbq$\emph{-trivial} pairs, which are at the heart of the relations between the Galois group and the triviality of the exponent lattice of a polynomial, are characterized. An effective algorithm is developed to recognize these pairs. Based on this, a new algorithm is designed to prove the triviality of the exponent lattice of a generic irreducible polynomial, which considerably improves a state-of-the-art algorithm of the same type when the polynomial degree becomes larger. In addition, the concept of the Galois-like groups of a polynomial is introduced. Some properties of the Galois-like groups are proved and, more importantly, a sufficient and necessary condition is given for a polynomial (which is not necessarily irreducible) to have trivial exponent lattice.
