# QthPower

## Description

This package computes the integral closure of type I affine domains and some slightly more general integral extensions of polynomial rings P using the qth-power algorithm from D.A.Leonard, Finding the missing functions for one-point AG codes, IEEE Trans. Inform.Theory, 47(6), 2001, pp. 2566-3573, D.A.Leonard and R.Pellikaan, Integral closures and weight functions over finite fields, Finite Fields and Their Applications 9(4), 2003, pp. 479-504, D.A.Leonard, A weighted module view of integral closures of affine domains of type I, Advances in Mathematics of Communication 3(1), 2009, pp. 1-11. ({ t icFracP} in the { t IntegralClosure} package of { t Macaulay2} and { t normalP} in { t Singular}'s { t normal} package are attempts to generalize this to generic input by ignoring all of the structure that is required by this package.) Also this package contains the extension to examples over the rationals; which, in turn, allows for quicker answers over ZZ/q for most large q, which can be produced if desired merely by changing the coefficient ring from QQ to ZZ/q.

## Version

This documentation describes version 1.02 of QthPower.

## Source code

The source code from which this documentation is derived is in the file QthPower.m2.

## Exports

• Functions and commands
• grevlexWeight -- transform a weight matrix into a monomial ordering matrix
• minimization -- change to a better Noether normalization suggested by the induced weights
• qthConductor -- computes a conductor element which also lives in the given Noether normalization, P
• qthIntegralClosure -- computes integral closures in positive characteristic
• rationalIntegralClosure -- computes integral closures over the rationals
• testWeightMatrix -- test compatibility of weight matrix with Groebner basis elements
• weightGrevlex -- transform a weight matrix into a monomial ordering matrix
• Methods
• "grevlexWeight(Matrix)" -- see grevlexWeight -- transform a weight matrix into a monomial ordering matrix
• "minimization(List,List,Ring,Matrix)" -- see minimization -- change to a better Noether normalization suggested by the induced weights
• "qthConductor(Ideal,ZZ)" -- see qthConductor -- computes a conductor element which also lives in the given Noether normalization, P
• "qthIntegralClosure(Matrix,Ring,List)" -- see qthIntegralClosure -- computes integral closures in positive characteristic
• "rationalIntegralClosure(Matrix,Ring,List)" -- see rationalIntegralClosure -- computes integral closures over the rationals
• "testWeightMatrix(Matrix,Ring,List)" -- see testWeightMatrix -- test compatibility of weight matrix with Groebner basis elements
• "weightGrevlex(Matrix)" -- see weightGrevlex -- transform a weight matrix into a monomial ordering matrix

## For the programmer

The object QthPower is .