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A New Program for Computing the P-Linear System Cardinality that Determines the Group of Weil Divisors of a Zariski Surface

C. L. Rogers
Journal / Anthology

Master's thesis, University of Arkansas at Little Rock
Year: 1995

Previously an algorithm and associated computer program for determining the group of Weil divisors of a normal Zariski surface, Xg, given by zp=g(x,y), where p>0 is the characteristic of a fixed algebraically closed field, k, containing g(x,y), has been presented. The algorithm generates a p-linear system of equations that has a set of solutions isomorphic to the divisor class group of the surface. The cardinality of this solution set completely determines the class group. The divisor class group, which is an abelian group and a geometric invariant, assists in the classification of algebraic surfaces over a fixed algebraically closed field. In this paper we describe a new algorithm (and computer program) that is more general and efficient than the previous version. This new algorithm has applications to general systems of p-linear equations of degree p. The development time was reduced while shortening the number of lines of programming (approximately 10:1) using the computer algebra system, Mathematica, and removing restrictions on problem size. Problems with p values up to 17 were calculated and compared with known values for divisor class groups of these types as well as the examples in Blass.

*Mathematics > Algebra > Field and Ring Theory