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Introduction

This chapter deals with pseudo-matrices. We follow the common terminology and conventions introduced in Henri Cohen (2000), however, we operate on rows, not on columns.

Let be a Dedekind domain, typically, the maximal order of some number field , further fix some finite dimensional -vectorspace (with some basis), frequently or the -structure of some extension of . Since in general is not a PID, the -modules in are usually not free, but still projective.

Any finitely generated -module can be represented as a pseudo-matrix PMat as follows: The structure theory of -modules gives the existence of (fractional) -ideals and elements such that and the sum is direct.

Following Cohen we call modules of the form for some ideal and a pseudo element. A system is called a pseudo-generating system for if . A pseudo-generating system is called a pseudo-basis if the are -linear independent.

A pseudo-matrix is a tuple containing a vector of ideals () and a matrix . The -th row together with the -th ideal defines a pseudo-element, thus an -module, all of them together generate a module .

A pseudo-matrix is said to be in pseudo-hnf if is essentially upper triangular. Similar to the classical hnf, there is an algorithm that transforms any pseudo-matrix into one in pseudo-hnf while maintaining the module.

Creation

In general to create a PMat one has to specify a matrix and a vector of ideals:

# pseudo_matrixMethod.

pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}

Returns the (row) pseudo matrix representing the -module where are the ideals in and the rows of .

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# pseudo_matrixMethod.

pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}

Returns the (row) pseudo matrix representing the -module where are the ideals in and the rows of .

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# pseudo_matrixMethod.

pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}

Returns the free (row) pseudo matrix representing the -module where are the rows of .

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(Those functions are also available as pseudo_matrix)

Operations

# coefficient_idealsMethod.

coefficient_ideals(M::PMat)

Returns the vector of coefficient ideals.

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# matrixMethod.

matrix(M::PMat)

Returns the matrix part of the PMat.

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# base_ringMethod.

base_ring(M::PMat)

The PMat defines an -module for some maximal order . This function returns the that was used to defined .

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# pseudo_hnfMethod.

pseudo_hnf(P::PMat)

Transforms into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module.

A optional second argument can be specified as a symbols, indicating the desired shape of the echelon form. Possible are :upperright (the default) and :lowerleft

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# pseudo_hnf_with_transformMethod.

pseudo_hnf_with_transform(P::PMat)

Transforms into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module. The used transformation is returned as a second return value.

A optional second argument can be specified as a symbol, indicating the desired shape of the echelon form. Possible are :upperright (the default) and :lowerleft

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Examples