<!-- 
  HAPPRIME - functions.xml.in
  Function documentation template file for HAPprime
  Paul Smith

  Copyright (C)  2007-2008
  Paul Smith
  National University of Ireland Galway

  $Id: functions.xml.in 200 2008-02-05 14:29:47Z pas $
-->

<!-- ********************************************************** -->
<Chapter>
  <Heading>General Functions</Heading>

  Some of the functions provided by &HAPprime; are not specifically aimed
  at homological algebra or extending the &HAP; package. The functions
  in this chapter, which are used internally by &HAPprime; extend some of the 
  standard &GAP; functions and datatypes.

  <Section>
    <Heading>Matrices</Heading>

    See <Ref Chap="Matrices" BookName="ref"/> in the
    &GAP; reference manual for the standard &GAP; matrix functions. 

    <#Include Label="SumIntersectionMatDestructive_manBaseMat">
    <#Include Label="SolutionMat_manBaseMat">
    <#Include Label="IsSameSubspace_manBaseMat">
    <#Include Label="PrintDimensionsMat_manBaseMat">

  </Section>

  <Section>
    <Heading>Polynomials</Heading>

    See <Ref Chap="Polynomials and Rational Functions" BookName="ref"/> in 
    the &GAP; reference manual for the functions provided by &GAP; for 
    representing an manipulating polynomials.

    <#Include Label="TermsOfPolynomial_manPolynomial">
    <#Include Label="UnivariateMonomialsOfMonomial_manPolynomial">
    <#Include Label="IndeterminateAndCoefficientOfUnivariateMonomial_manPolynomial">
    <#Include Label="ReduceIdeal_manPolynomial">
    <#Include Label="ReducePolynomialRingPresentation_manPolynomial">

    <Subsection>
      <Heading>Examples</Heading>
    
<Example><![CDATA[
gap> ring := PolynomialRing(Integers, 2);;
gap> i := IndeterminatesOfPolynomialRing(ring);;
gap> poly := i[1] + i[1]*i[2]^2 + 3*i[2]^3;
x_1*x_2^2+3*x_2^3+x_1
gap> terms := TermsOfPolynomial(poly);
[ [ x_1, 1 ], [ x_2^3, 3 ], [ x_1*x_2^2, 1 ] ]
gap> UnivariateMonomialsOfMonomial(terms[3][1]);
[ x_1, x_2^2 ]
gap> IndeterminateAndCoefficientOfUnivariateMonomial(last[2]);
[ x_2, 2 ]
]]></Example>

<Example><![CDATA[
gap> ring := PolynomialRing(GF(2), 2);;
gap> i := IndeterminatesOfPolynomialRing(ring);;
gap> I := [i[1]^2 + i[2], i[1]^3 + i[2]^3];
[ x_1^2+x_2, x_1^3+x_2^3 ]
gap> ReduceIdeal(I, MonomialLexOrdering());
[ x_1^2+x_2, x_2^3+x_1*x_2 ]
]]></Example>

<Example><![CDATA[
gap> ring := PolynomialRing(GF(2), 3);;
gap> i := IndeterminatesOfPolynomialRing(ring);;
gap> ideal := [ i[3]^2 + i[1] + i[2] ];
[ x_3^2+x_1+x_2 ]
gap> ReducePolynomialRingPresentation(ring, ideal);
[ GF(2)[x_1,x_3], [  ] ]
]]></Example>
    </Subsection>
  </Section>


</Chapter>