Allan K. Steel

Some Algorithms in Invariant Theory of Finite Groups

We present algorithms which calculate the invariant ring K[V] G of a finite group G. Our focus of interest lies on the modular case, i.e., the case where |G| is divided by the characteristic of K. We give easy algorithms to compute several interesting properties of the invariant ring, such as the Cohen-Macaulay property, depth, the β-number and syzygies.

Conquering inseparability: Primary decomposition and multivariate factorization over algebraic function fields of positive characteristic

Algebraic function fields of positive characteristic are non-perfect fields, and many standard algorithms for solving some fundamental problems in commutative algebra simply do not work over these fields. This paper presents practical algorithms for the first time for (1) computing the primary decomposition of ideals of polynomial rings defined over such fields and (2) factoring arbitrary multivariate polynomials over such fields. Difficulties involving inseparability and the situation where the transcendence degree is greater than one are completely overcome, while the algorithms avoid explicit construction of any extension of the input base field. As a corollary, the problem of computing the primary decomposition of a positive-dimensional ideal over a finite field is also solved. The algorithms perform very effectively in an implementation within the Magma Computer Algebra System, and an analysis of their practical performance is given.

A new algorithm for the computation of canonical forms of matrices over fields

Abstract A new algorithm is presented for the computation of canonical forms of matrices over fields. These are the Primary Rational , Rational , and Jordan canonical forms. The algorithm works by obtaining a decomposition of the vector space acted on by the given matrix into primary cyclic spaces (spaces whose minimal polynomials with respect to the matrix are powers of irreducible polynomials). An efficient implementation of the algorithm is incorporated in the MAGMA Computer Algebra System.

Lattices of compatibly embedded finite fields

Abstract The design of a computational facility for finite fields that allows complete freedom in the manner in which fields are constructed, is complicated by the fact that a field of fixed isomorphism type K may be constructed in many different ways. It is desirable that the user be able to perform simultaneous computations in different versions of K in such a way that isomorphisms identifying elements in the different versions are applied automatically whenever necessary. This paper presents a coherent scheme for solving this problem based on an efficient method for compatibly embedding one field within another. This scheme forms a central component of the MAGMA module for finite fields. The paper also outlines the different representations of finite fields employed in the package and comments briefly on some of the major algorithms.

A New Scheme for Computing with Algebraically Closed Fields

A new scheme is presented for computing with an algebraic closure of the rational field.It avoids factorization of polynomials over extension fields, but gives the illusion of a genuine field to the user.A technique of modular evaluation into a finite field ensures that a unique genuine field is simulated by the scheme and also provides fast optimizations for some critical operations.F ast modular matrix techniques are also used for several non-trivial operations.The scheme has been successfully implemented within the Magma Computer Algebra System.

Computing with algebraically closed fields

A practical computational system is described for computing with an algebraic closure of a field. The system avoids factorization of polynomials over extension fields, but gives the illusion of a genuine field to the user. All roots of an arbitrary polynomial defined over such an algebraically closed field can be constructed and are easily distinguished within the system. The difficult case of inseparable extensions of function fields of positive characteristic is also handled properly by the system. A technique of modular evaluation into a finite field critically ensures that a unique genuine field is simulated by the system but also provides fast optimizations for some fundamental operations. Fast matrix techniques are also used for several non-trivial operations. The system has been successfully implemented within the Magma Computer Algebra System, and several examples are presented, using this implementation.

Computing subgroups of bounded index in a finite group

We describe a practical algorithm for computing representatives of the conjugacy classes of subgroups up to a given index in a finite group. This algorithm has been implemented in Magma, and we present some performance statistics.