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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<?xml version="1.0" encoding="UTF-8"?>

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<!-- This is an automatically generated file. -->

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<Chapter Label="Chapter_Examples">

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<Heading>Examples</Heading>

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<P/>

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<Section Label="Chapter_Examples_Section_Generators">

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<Heading>Generators</Heading>

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<P/>

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<Example><![CDATA[

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gap> C := NmzCone(["integral_closure",[[2,1],[1,3]]]);

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<a Normaliz cone>

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gap> NmzHasConeProperty(C,"HilbertBasis");

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false

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gap> NmzHasConeProperty(C,"SupportHyperplanes");

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false

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gap> NmzConeProperty(C,"HilbertBasis");

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[ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 1 ] ]

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gap> NmzHasConeProperty(C,"SupportHyperplanes");

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true

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gap> NmzConeProperty(C,"SupportHyperplanes");

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[ [ -1, 2 ], [ 3, -1 ] ]

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]]></Example>

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<P/>

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</Section>

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<Section Label="Chapter_Examples_Section_System_of_equations">

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<Heading>System of equations</Heading>

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<Example><![CDATA[

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gap> D := NmzCone(["equations",[[1,2,-3]], "grading",[[0,-1,3]]]);

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<a Normaliz cone>

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gap> NmzCompute(D,["DualMode","HilbertSeries"]);

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true

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gap> NmzHilbertBasis(D);

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[ [ 1, 1, 1 ], [ 0, 3, 2 ], [ 3, 0, 1 ] ]

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gap> NmzHilbertSeries(D);

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[ t^2-t+1, [ [ 1, 1 ], [ 3, 1 ] ] ]

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gap> NmzHasConeProperty(D,"SupportHyperplanes");

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true

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gap> NmzSupportHyperplanes(D);

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[ [ 1, 0, 0 ], [ 1, 3, -3 ] ]

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gap> NmzEquations(D);

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[ [ 1, 2, -3 ] ]

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]]></Example>

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<P/>

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</Section>

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<Section Label="Chapter_Examples_Section_System_of_inhomogeneous_equations">

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<Heading>System of inhomogeneous equations</Heading>

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<Example><![CDATA[

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gap> P := NmzCone(["inhom_equations",[[1,2,-3,1]], "grading", [[1,1,1]]]);

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<a Normaliz cone>

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gap> NmzIsInhomogeneous(C);

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false

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gap> NmzIsInhomogeneous(P);

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true

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gap> NmzHilbertBasis(P);

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[ [ 1, 1, 1, 0 ], [ 3, 0, 1, 0 ], [ 0, 3, 2, 0 ] ]

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gap> NmzModuleGenerators(P);

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[ [ 0, 1, 1, 1 ], [ 2, 0, 1, 1 ] ]

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]]></Example>

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<P/>

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</Section>

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<Section Label="Chapter_Examples_Section_Combined_input">

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<Heading>Combined input</Heading>

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Normaliz also allows the combination of different kinds of input, e.g.

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multiple constraint types, or additional data like a grading.

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<P/>

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Suppose that you have a 3 by 3 <Q>square</Q> of nonnegative integers such that

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the 3 numbers in all rows, all columns, and both diagonals sum to the same

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constant <M>M</M>. Sometimes such squares are called magic and <M>M</M> is the

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magic constant. This leads to a linear system of equations. The magic constant

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is a natural choice for the grading. Additionally we force here the 4 corner

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to have even value by adding congruences.

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<P/>

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<Example><![CDATA[

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gap> Magic3x3even := NmzCone(["equations",

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> [ [1, 1, 1,  -1, -1, -1,   0,  0,  0],

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>   [1, 1, 1,   0,  0,  0,  -1, -1, -1],

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>   [0, 1, 1,  -1,  0,  0,  -1,  0,  0],

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>   [1, 0, 1,   0, -1,  0,   0, -1,  0],

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>   [1, 1, 0,   0,  0, -1,   0,  0, -1],

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>   [0, 1, 1,   0, -1,  0,   0,  0, -1],

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>   [1, 1, 0,   0, -1,  0,  -1,  0,  0] ],

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> "congruences",

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> [ [1, 0, 0,   0, 0, 0,   0, 0, 0,  2],

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>   [0, 0, 1,   0, 0, 0,   0, 0, 0,  2],

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>   [0, 0, 0,   0, 0, 0,   1, 0, 0,  2],

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>   [0, 0, 0,   0, 0, 0,   0, 0, 1,  2] ],

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> "grading",

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> [ [1, 1, 1,   0, 0, 0,   0, 0, 0] ] ] );

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<a Normaliz cone>

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gap> NmzHilbertBasis(Magic3x3even);

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[ [ 0, 4, 2, 4, 2, 0, 2, 0, 4 ], [ 2, 0, 4, 4, 2, 0, 0, 4, 2 ],

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  [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 4, 0, 0, 2, 4, 4, 0, 2 ],

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  [ 4, 0, 2, 0, 2, 4, 2, 4, 0 ], [ 2, 3, 4, 5, 3, 1, 2, 3, 4 ],

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  [ 2, 5, 2, 3, 3, 3, 4, 1, 4 ], [ 4, 1, 4, 3, 3, 3, 2, 5, 2 ],

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  [ 4, 3, 2, 1, 3, 5, 4, 3, 2 ] ]

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gap> NmzHilbertSeries(Magic3x3even);

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[ t^3+3*t^2-t+1, [ [ 1, 1 ], [ 2, 2 ] ] ]

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gap> NmzHilbertQuasiPolynomial(Magic3x3even);

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[ 1/2*t^2+t+1, 1/2*t^2-1/2 ]

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]]></Example>

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<P/>

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</Section>

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<Section Label="Chapter_Examples_Section_Using_the_dual_mode">

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<Heading>Using the dual mode</Heading>

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For solving systems of equations and inequalities it is often faster to use

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the dual Normaliz algorithm. We demonstrate how to use it with an

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inhomogeneous system of equations and inequalities.

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<P/>

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The input consists of a system of 8 inhomogeneous equations in R^3. The first

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row of the matrix M encodes the inequality <M>8x + 8y + 8z + 7 \geq 0</M>.

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Additionally we say that <M>x, y, z</M> should be non-negative by giving the

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sign vector and use the total degree.

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<Example><![CDATA[

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gap> M := [

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>  [ 8,  8,  8,  7 ],

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>  [ 0,  4,  0,  1 ],

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>  [ 0,  1,  0,  7 ],

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>  [ 0, -2,  0,  7 ],

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>  [ 0, -2,  0,  1 ],

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>  [ 8, 48,  8, 17 ],

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>  [ 1,  6,  1, 34 ],

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>  [ 2,-12, -2, 37 ],

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>  [ 4,-24, -4, 14 ]

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> ];

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[ [ 8, 8, 8, 7 ], [ 0, 4, 0, 1 ], [ 0, 1, 0, 7 ], [ 0, -2, 0, 7 ],

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  [ 0, -2, 0, 1 ], [ 8, 48, 8, 17 ], [ 1, 6, 1, 34 ],

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  [ 2, -12, -2, 37 ], [ 4, -24, -4, 14 ] ]

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gap> D := NmzCone(["inhom_inequalities", M,

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>               "signs", [[1,1,1]],

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>               "grading", [[1,1,1]]]);

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<a Normaliz cone>

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gap> NmzCompute(D,["DualMode","HilbertBasis","ModuleGenerators"]);

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true

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gap> NmzHilbertBasis(D);

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[ [ 1, 0, 0, 0 ], [ 1, 0, 1, 0 ] ]

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gap> NmzModuleGenerators(D);

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[ [ 0, 0, 0, 1 ], [ 0, 0, 1, 1 ], [ 0, 0, 2, 1 ], [ 0, 0, 3, 1 ] ]

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]]></Example>

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<P/>

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As result we get the Hilbert basis of the cone of the solutions to the

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homogeneous system and the module generators which are the base

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solutions to the inhomogeneous system.

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<P/>

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</Section>

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</Chapter>

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