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<p><a id="X7DFB63A97E67C0A1" name="X7DFB63A97E67C0A1"></a></p>
<div class="ChapSects"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7B818AF0834AE510">1.1 <span class="Heading">Ring Constructions for Supported External Computer Algebra Systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X827FA5FD7CBEE73C">1.1-1 <span class="Heading">external <strong class="pkg">GAP</strong></span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X840A7A5D7AE407F2">1.1-2 <span class="Heading"><strong class="pkg">Singular</strong></span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X839147CF813312D6">1.1-3 <span class="Heading"><strong class="pkg">MAGMA</strong></span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83D66E0279CF39E6">1.1-4 <span class="Heading"><strong class="pkg">Macaulay2</strong></span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A9D29E187AA3F83">1.1-5 <span class="Heading"><strong class="pkg">Sage</strong></span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83E0B3C584764593">1.1-6 <span class="Heading"><strong class="pkg">Maple</strong></span></a>
</span>
</div></div>
</div>
<h3>1 <span class="Heading">Introduction</span></h3>
<p>This package is part of the <strong class="pkg">homalg</strong> project <a href="chapBib_mj.html#biBhomalg-project">[tpa22]</a>. The role of the package is described in the manual of the <strong class="pkg">homalg</strong> package.</p>
<p><a id="X7B818AF0834AE510" name="X7B818AF0834AE510"></a></p>
<h4>1.1 <span class="Heading">Ring Constructions for Supported External Computer Algebra Systems</span></h4>
<p>Here are some of the supported ring constructions:</p>
<p><a id="X827FA5FD7CBEE73C" name="X827FA5FD7CBEE73C"></a></p>
<h5>1.1-1 <span class="Heading">external <strong class="pkg">GAP</strong></span></h5>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegersInExternalGAP( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zz );</span>
<An external ring residing in the CAS GAP>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2 := HomalgRingOfIntegersInExternalGAP( 2, zz );</span>
GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2 );</span>
<An external ring residing in the CAS GAP>
</pre></div>
<p><code class="code">F2 := HomalgRingOfIntegersInExternalGAP( 2 )</code> would launch another GAP.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Z4 := HomalgRingOfIntegersInExternalGAP( 4, zz );</span>
Z/4Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Z4 );</span>
<An external ring residing in the CAS GAP>
<span class="GAPprompt">gap></span> <span class="GAPinput">Z_4 := HomalgRingOfIntegersInExternalGAP( zz ) / 4;</span>
Z/( 4 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Z_4 );</span>
<A residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := HomalgFieldOfRationalsInExternalGAP( zz );</span>
Q
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Q );</span>
<An external ring residing in the CAS GAP>
</pre></div>
<p><a id="X840A7A5D7AE407F2" name="X840A7A5D7AE407F2"></a></p>
<h5>1.1-2 <span class="Heading"><strong class="pkg">Singular</strong></span></h5>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Singular ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">F2 := HomalgRingOfIntegersInSingular( 2 );</span>
GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2 );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2s := HomalgRingOfIntegersInSingular( 2, "s" ,F2 );</span>
GF(2)(s)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2s );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegersInSingular( F2 );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := HomalgFieldOfRationalsInSingular( F2 );</span>
Q
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Q );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qs := HomalgFieldOfRationalsInSingular( "s", F2 );</span>
Q(s)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qs );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qi := HomalgFieldOfRationalsInSingular( "i", "i^2+1", Q );</span>
Q[i]/(i^2+1)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qi );</span>
<An external ring residing in the CAS Singular>
#@fi
</pre></div>
<p><code class="code">Q := HomalgFieldOfRationalsInSingular( )</code> would launch another Singular.</p>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Singular ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">F2xyz := F2 * "x,y,z";</span>
GF(2)[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2xyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2sxyz := F2s * "x,y,z";</span>
GF(2)(s)[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2sxyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2xyzw := F2xyz * "w";</span>
GF(2)[x,y,z][w]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2xyzw );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2sxyzw := F2sxyz * "w";</span>
GF(2)(s)[x,y,z][w]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2sxyzw );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">zzxyz := zz * "x,y,z";</span>
Z[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zzxyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">zzxyzw := zzxyz * "w";</span>
Z[x,y,z][w]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zzxyzw );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := Q * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qxyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qsxyz := Qs * "x,y,z";</span>
Q(s)[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qsxyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qixyz := Qi * "x,y,z";</span>
(Q[i]/(i^2+1))[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qixyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyzw := Qxyz * "w";</span>
Q[x,y,z][w]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qxyzw );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qsxyzw := Qsxyz * "w";</span>
Q(s)[x,y,z][w]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qsxyzw );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );</span>
Q[x,y,z]<Dx,Dy,Dz>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Dxyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Exyz := ExteriorRing( Qxyz, "e,f,g" );</span>
Q{e,f,g}
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Exyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dsxyz := RingOfDerivations( Qsxyz, "Dx,Dy,Dz" );</span>
Q(s)[x,y,z]<Dx,Dy,Dz>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Dsxyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Esxyz := ExteriorRing( Qsxyz, "e,f,g" );</span>
Q(s){e,f,g}
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Esxyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dixyz := RingOfDerivations( Qixyz, "Dx,Dy,Dz" );</span>
(Q[i]/(i^2+1))[x,y,z]<Dx,Dy,Dz>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Dixyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">Eixyz := ExteriorRing( Qixyz, "e,f,g" );</span>
(Q[i]/(i^2+1)){e,f,g}
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Eixyz );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">qring := HomalgQRingInSingular( Qxyz, "x*y" );</span>
Q[x,y,z]/( x*y )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( qring );</span>
<An external ring residing in the CAS Singular>
<span class="GAPprompt">gap></span> <span class="GAPinput">"z + x*y" / qring = "z" / qring;</span>
true
#@fi
</pre></div>
<p><a id="X839147CF813312D6" name="X839147CF813312D6"></a></p>
<h5>1.1-3 <span class="Heading"><strong class="pkg">MAGMA</strong></span></h5>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_MAGMA ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegersInMAGMA( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zz );</span>
<An external ring residing in the CAS MAGMA>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2 := HomalgRingOfIntegersInMAGMA( 2, zz );</span>
GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2 );</span>
<An external ring residing in the CAS MAGMA>
#@fi
</pre></div>
<p><code class="code">F2 := HomalgRingOfIntegersInMAGMA( 2 )</code> would launch another MAGMA.</p>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_MAGMA ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">Z_4 := HomalgRingOfIntegersInMAGMA( zz ) / 4;</span>
Z/( 4 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Z_4 );</span>
<A residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := HomalgFieldOfRationalsInMAGMA( zz );</span>
Q
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Q );</span>
<An external ring residing in the CAS MAGMA>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2xyz := F2 * "x,y,z";</span>
GF(2)[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2xyz );</span>
<An external ring residing in the CAS MAGMA>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := Q * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qxyz );</span>
<An external ring residing in the CAS MAGMA>
<span class="GAPprompt">gap></span> <span class="GAPinput">Exyz := ExteriorRing( Qxyz, "e,f,g" );</span>
Q{e,f,g}
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Exyz );</span>
<An external ring residing in the CAS MAGMA>
#@fi
</pre></div>
<p><a id="X83D66E0279CF39E6" name="X83D66E0279CF39E6"></a></p>
<h5>1.1-4 <span class="Heading"><strong class="pkg">Macaulay2</strong></span></h5>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Macaulay2 ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegersInMacaulay2( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zz );</span>
<An external ring residing in the CAS Macaulay2>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2 := HomalgRingOfIntegersInMacaulay2( 2, zz );</span>
GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2 );</span>
<An external ring residing in the CAS Macaulay2>
#@fi
</pre></div>
<p><code class="code">F2 := HomalgRingOfIntegersInMacaulay2( 2 )</code> would launch another Macaulay2.</p>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Macaulay2 ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">Z_4 := HomalgRingOfIntegersInMacaulay2( zz ) / 4;</span>
Z/( 4 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Z_4 );</span>
<A residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := HomalgFieldOfRationalsInMacaulay2( zz );</span>
Q
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Q );</span>
<An external ring residing in the CAS Macaulay2>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2xyz := F2 * "x,y,z";</span>
GF(2)[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2xyz );</span>
<An external ring residing in the CAS Macaulay2>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := Q * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qxyz );</span>
<An external ring residing in the CAS Macaulay2>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );</span>
Q[x,y,z]<Dx,Dy,Dz>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Dxyz );</span>
<An external ring residing in the CAS Macaulay2>
<span class="GAPprompt">gap></span> <span class="GAPinput">Exyz := ExteriorRing( Qxyz, "e,f,g" );</span>
Q{e,f,g}
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Exyz );</span>
<An external ring residing in the CAS Macaulay2>
#@fi
</pre></div>
<p><a id="X7A9D29E187AA3F83" name="X7A9D29E187AA3F83"></a></p>
<h5>1.1-5 <span class="Heading"><strong class="pkg">Sage</strong></span></h5>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Sage ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegersInSage( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zz );</span>
<An external ring residing in the CAS Sage>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2 := HomalgRingOfIntegersInSage( 2, zz );</span>
GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2 );</span>
<An external ring residing in the CAS Sage>
#@fi
</pre></div>
<p><code class="code">F2 := HomalgRingOfIntegersInSage( 2 )</code> would launch another Sage.</p>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Sage ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">Z_4 := HomalgRingOfIntegersInSage( zz ) / 4;</span>
Z/( 4 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Z_4 );</span>
<A residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := HomalgFieldOfRationalsInSage( zz );</span>
Q
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Q );</span>
<An external ring residing in the CAS Sage>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2x := F2 * "x";</span>
GF(2)[x]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2x );</span>
<An external ring residing in the CAS Sage>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qx := Q * "x";</span>
Q[x]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qx );</span>
<An external ring residing in the CAS Sage>
#@fi
</pre></div>
<p><a id="X83E0B3C584764593" name="X83E0B3C584764593"></a></p>
<h5>1.1-6 <span class="Heading"><strong class="pkg">Maple</strong></span></h5>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Maple ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegersInMaple( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zz );</span>
<An external ring residing in the CAS Maple>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2 := HomalgRingOfIntegersInMaple( 2, zz );</span>
GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2 );</span>
<An external ring residing in the CAS Maple>
#@fi
</pre></div>
<p><code class="code">F2 := HomalgRingOfIntegersInMaple( 2 )</code> would launch another Maple.</p>
<div class="example"><pre>
#@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Maple ).stdout )
<span class="GAPprompt">gap></span> <span class="GAPinput">Z4 := HomalgRingOfIntegersInMaple( 4, zz );</span>
Z/4Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Z4 );</span>
<An external ring residing in the CAS Maple>
<span class="GAPprompt">gap></span> <span class="GAPinput">Z_4 := HomalgRingOfIntegersInMaple( zz ) / 4;</span>
Z/( 4 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Z_4 );</span>
<A residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := HomalgFieldOfRationalsInMaple( zz );</span>
Q
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Q );</span>
<An external ring residing in the CAS Maple>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2xyz := F2 * "x,y,z";</span>
GF(2)[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( F2xyz );</span>
<An external ring residing in the CAS Maple>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := Q * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Qxyz );</span>
<An external ring residing in the CAS Maple>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );</span>
Q[x,y,z]<Dx,Dy,Dz>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Dxyz );</span>
<An external ring residing in the CAS Maple>
<span class="GAPprompt">gap></span> <span class="GAPinput">Exyz := ExteriorRing( Qxyz, "e,f,g" );</span>
Q{e,f,g}
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Exyz );</span>
<An external ring residing in the CAS Maple>
#@fi
</pre></div>
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