Invariants of Finite Groups Acting as Flag Automorphisms

Let K be a field and suppose that G is a finite group that acts faithfully on $(x1,...,xm) by automorphisms of the form g(xi)=ai(g)xi+bi(g), where ai(g),bi(g) \in K(x1,...,xi-1) for all g \in G and all i=1,...,m. As shown by Miyata, the fixed field K(x1,...,xi-1)G is purely transcendental over K and admits a transcendence basis {\phi1,...,\phim}, where \phii is in K(x1,...,xi-1) [xi]G and has minimal positive degree di in xi. We determine exactly the degree di of each invariant \phii as a polynomial in xi and show the relation d1 ... dm=|G|. As an application, we compute a generic polynomial for the dihedral group D8 of order 16 in characteristic 2

Generic Polynomials for Transitive Permutation Groups of Degree 8 and 9

We compute generic polynomials for certain transitive permutation groups of degree 8 and 9, namely SL(2,3), the generalized dihedral group: C2 \ltimes (C3 x C3), and the Iwasawa group of order 16: M16. Rikuna proves the existence of a generic polynomial for SL(2,3) in four parameters; we extend a computation of Grobner to give an alternative proof of existence for this group\u27s generic polynomial. We establish that the generic dimension and essential dimension of the generalized dihedral group are two. We establish over the rationals that the generic dimension and essential dimension of SL(2,3) and M16 are four

Implementation of prime decomposition of polynomial ideals over small finite fields

AbstractAn algorithm for the prime decomposition of polynomial ideals over small finite fields is proposed and implemented on the basis of previous work of the second author. To achieve better performance, several improvements are added to the existing algorithm, with strategies for computational flow proposed, based on experimental results. The practicality of the algorithm is examined by testing the implementation experimentally, which also reveals information about the quality of the implementation

Implementation of prime decomposition of polynomial ideals over small finite fields

AbstractAn algorithm for the prime decomposition of polynomial ideals over small finite fields is proposed and implemented on the basis of previous work of the second author. To achieve better performance, several improvements are added to the existing algorithm, with strategies for computational flow proposed, based on experimental results. The practicality of the algorithm is examined by testing the implementation experimentally, which also reveals information about the quality of the implementation

On the field intersection problem of quartic generic polynomials via formal Tschirnhausen transformation

Let $k$ be a field of characteristic $\neq 2$. We give an answer to the field intersection problem of quartic generic polynomials over $k$ via formal Tschirnhausen transformation and multi-resolvent polynomials.Comment: 32 pages, 6 tables, to appear in Comment. Math. Univ. St. Paul