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<head><title>flattenRing -- write a ring as a (quotient) of a polynomial ring over ZZ or a prime field</title>
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<div><h1>flattenRing -- write a ring as a (quotient) of a polynomial ring over ZZ or a prime field</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>(S,F) = flattenRing R</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span> or <span>an <a href="___Ideal.html">ideal</a></span></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>S</tt>, <span>a <a href="___Ring.html">ring</a></span> or <span>an <a href="___Ideal.html">ideal</a></span>a ring isomorphic to the original ring, flattened in the sense that it is a quotient ring of a polyonial ring over the bottom-most coefficient ring; or in case an ideal was provided, the corresponding ideal</span></li>
<li><span><tt>F</tt>, <span>a <a href="___Ring__Map.html">ring map</a></span>, the isomorphism from <tt>R</tt> to <tt>S</tt></span></li>
</ul>
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</li>
<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>CoefficientRing => </tt><span><span>a <a href="___Ring.html">ring</a></span>, <span>default value null</span>, the desired coefficient ring for the result</span></span></li>
<li><span><tt>Result => </tt><span><span>default value (Thing,RingMap)</span>, the number or type(s) of result(s) desired. Three possible results are available: an ideal (<a href="___Ideal.html" title="the class of all ideals">Ideal</a>) or the corresponding quotient ring (<a href="___Ring.html" title="the class of all rings">Ring</a>), the isomorphism from <tt>R</tt> to the flattened ring (<a href="___Ring__Map.html" title="the class of all ring maps">RingMap</a>), and the inverse isomorphism (<a href="___Ring__Map.html" title="the class of all ring maps">RingMap</a>). Asking for a result of type <a href="___Nothing.html" title="the empty class">Nothing</a> will yield <a href="_null.html" title="the unique member of the empty class">null</a> in the corresponding position. Omitting the result type but leaving its comma will yield the default.</span></span></li>
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<div class="single"><h2>Description</h2>
<div><p>If the optional argument is not given, then the coefficient ring of the result is either <a href="___Z__Z.html" title="the class of all integers">ZZ</a> or the base field.</p>
<p>The inverse of the isomorphism <tt>F</tt> is obtainable with <tt>F^-1</tt>.</p>
<table class="examples"><tr><td><pre>i1 : A = ZZ[a]/(a^2-3)
o1 = A
o1 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i2 : B = A[x,y,z]/(a*x^2-y^2-z^2, y^3, z^3)
o2 = B
o2 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i3 : (D,F) = flattenRing B;</pre>
</td></tr>
<tr><td><pre>i4 : F
o4 = map(D,B,{x, y, z, a})
o4 : RingMap D <--- B</pre>
</td></tr>
<tr><td><pre>i5 : F^-1
o5 = map(B,D,{x, y, z, a})
o5 : RingMap B <--- D</pre>
</td></tr>
<tr><td><pre>i6 : D
o6 = D
o6 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i7 : describe D
ZZ[x, y, z, a]
o7 = -------------------------------
2 2 2 2 3 3
(a - 3, x a - y - z , y , z )</pre>
</td></tr>
<tr><td><pre>i8 : flattenRing(B,Result => Ideal)
2 2 2 2 3 3
o8 = ideal (a - 3, x a - y - z , y , z )
o8 : Ideal of ZZ[x, y, z, a]</pre>
</td></tr>
<tr><td><pre>i9 : flattenRing(B,Result => (Ideal,,))
2 2 2 2 3 3
o9 = (ideal (a - 3, x a - y - z , y , z ), map(ZZ[x, y, z, a],B,{x, y, z,
------------------------------------------------------------------------
a}), map(B,ZZ[x, y, z, a],{x, y, z, a}))
o9 : Sequence</pre>
</td></tr>
<tr><td><pre>i10 : flattenRing(B,Result => (,,))
ZZ[x, y, z, a]
o10 = (-------------------------------,
2 2 2 2 3 3
(a - 3, x a - y - z , y , z )
-----------------------------------------------------------------------
ZZ[x, y, z, a]
map(-------------------------------,B,{x, y, z, a}),
2 2 2 2 3 3
(a - 3, x a - y - z , y , z )
-----------------------------------------------------------------------
ZZ[x, y, z, a]
map(B,-------------------------------,{x, y, z, a}))
2 2 2 2 3 3
(a - 3, x a - y - z , y , z )
o10 : Sequence</pre>
</td></tr>
<tr><td><pre>i11 : flattenRing(B,Result => 3)
ZZ[x, y, z, a]
o11 = (-------------------------------,
2 2 2 2 3 3
(a - 3, x a - y - z , y , z )
-----------------------------------------------------------------------
ZZ[x, y, z, a]
map(-------------------------------,B,{x, y, z, a}),
2 2 2 2 3 3
(a - 3, x a - y - z , y , z )
-----------------------------------------------------------------------
ZZ[x, y, z, a]
map(B,-------------------------------,{x, y, z, a}))
2 2 2 2 3 3
(a - 3, x a - y - z , y , z )
o11 : Sequence</pre>
</td></tr>
<tr><td><pre>i12 : flattenRing(B,Result => (Nothing,Nothing,))
o12 = (, , map(B,ZZ[x, y, z, a],{x, y, z, a}))
o12 : Sequence</pre>
</td></tr>
</table>
<p>Warning: flattening the same ring with different options may yield a separately constructed rings, unequal to each other.</p>
<p>Flattening an ideal instead of a quotient ring can save a lot of time spent computing the Gröbner basis of the resulting ideal, if the flattened quotient is not needed.</p>
<table class="examples"><tr><td><pre>i13 : A = ZZ[a]/(a^2-3)
o13 = A
o13 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i14 : B = A[x,y,z]
o14 = B
o14 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i15 : J = ideal (a*x^2-y^2-z^2, y^3, z^3)
2 2 2 3 3
o15 = ideal (a*x - y - z , y , z )
o15 : Ideal of B</pre>
</td></tr>
<tr><td><pre>i16 : (J',F) = flattenRing J;</pre>
</td></tr>
<tr><td><pre>i17 : J'
2 2 2 2 3 3
o17 = ideal (a - 3, x a - y - z , y , z )
o17 : Ideal of ZZ[x, y, z, a]</pre>
</td></tr>
</table>
<p>In the following example, the coefficient ring of the result is the fraction field <tt>K</tt>.</p>
<table class="examples"><tr><td><pre>i18 : K = frac(ZZ[a])
o18 = K
o18 : FractionField</pre>
</td></tr>
<tr><td><pre>i19 : B = K[x,y,z]/(a*x^2-y^2-z^2, y^3, z^3)
o19 = B
o19 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i20 : (D,F) = flattenRing B
o20 = (B, map(B,B,{x, y, z, a}))
o20 : Sequence</pre>
</td></tr>
<tr><td><pre>i21 : describe D
K[x, y, z]
o21 = ------------------------
2 2 2 3 3
(a*x - y - z , y , z )</pre>
</td></tr>
</table>
<p>Once a ring has been declared to be a field with <a href="_to__Field_lp__Ring_rp.html" title="declare that a ring is a field">toField</a>, then it will be used as the coefficient ring.</p>
<table class="examples"><tr><td><pre>i22 : A = QQ[a]/(a^2-3);</pre>
</td></tr>
<tr><td><pre>i23 : L = toField A
o23 = L
o23 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i24 : B = L[x,y,z]/(a*x^2-y^2-z^2, y^3, z^3)
o24 = B
o24 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i25 : (D,F) = flattenRing(B[s,t])
o25 = (D, map(D,B[s, t],{s, t, x, y, z, a}))
o25 : Sequence</pre>
</td></tr>
<tr><td><pre>i26 : describe D
L[s, t, x, y, z]
o26 = ------------------------
2 2 2 3 3
(a*x - y - z , y , z )</pre>
</td></tr>
</table>
<p>If a larger coefficient ring is desired, use the optional CoefficientRing parameter.</p>
<table class="examples"><tr><td><pre>i27 : use L
o27 = L
o27 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i28 : C1 = L[s,t];</pre>
</td></tr>
<tr><td><pre>i29 : C2 = C1/(a*s-t^2);</pre>
</td></tr>
<tr><td><pre>i30 : C3 = C2[p_0..p_4]/(a*s*p_0)[q]/(q^2-a*p_1);</pre>
</td></tr>
<tr><td><pre>i31 : (D,F) = flattenRing(C3, CoefficientRing=>C2)
o31 = (D, map(D,C3,{q, p , p , p , p , p , s, t, a}))
0 1 2 3 4
o31 : Sequence</pre>
</td></tr>
<tr><td><pre>i32 : describe D
C2[q, p , p , p , p , p ]
0 1 2 3 4
o32 = -------------------------
2
(a*s*p , q - a*p )
0 1</pre>
</td></tr>
<tr><td><pre>i33 : (D,F) = flattenRing(C3, CoefficientRing=>QQ)
o33 = (D, map(D,C3,{q, p , p , p , p , p , s, t, a}))
0 1 2 3 4
o33 : Sequence</pre>
</td></tr>
<tr><td><pre>i34 : describe D
QQ[q, p , p , p , p , p , s, t, a]
0 1 2 3 4
o34 = -------------------------------------
2 2 2
(a - 3, - t + s*a, p s*a, q - p a)
0 1</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_presentation.html" title="presentation of a module or ring">presentation</a> -- presentation of a module or ring</span></li>
<li><span><a href="_coefficient__Ring.html" title="get the coefficient ring">coefficientRing</a> -- get the coefficient ring</span></li>
<li><span><a href="_describe.html" title="real description">describe</a> -- real description</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>flattenRing</tt> :</h2>
<ul><li>flattenRing(Ideal)</li>
<li>flattenRing(Ring)</li>
</ul>
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