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<head><title>frac -- construct a fraction field</title>
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<div><h1>frac -- construct a fraction 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>frac R</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span>, an integral domain</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Fraction__Field.html">fraction field</a></span>, the field of fractions of <tt>R</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : F = frac ZZ
o1 = QQ
o1 : Ring</pre>
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<tr><td><pre>i2 : F = frac (ZZ[a,b])
o2 = F
o2 : FractionField</pre>
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After invoking the <tt>frac</tt> command, the elements of the ring are treated as elements of the fraction field:<table class="examples"><tr><td><pre>i3 : R = ZZ/101[x,y];</pre>
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<tr><td><pre>i4 : gens gb ideal(x^2*y - y^3)
o4 = | x2y-y3 |
1 1
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : K = frac R;</pre>
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<tr><td><pre>i6 : gens gb ideal(x^2*y - y^3)
o6 = | 1 |
1 1
o6 : Matrix K <--- K</pre>
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Another way to obtain <tt>frac R</tt> is with <tt>x</tt><a href="__sl.html" title="a binary operator, usually used for division">/</a><tt>y</tt> where <tt>x, y</tt> are elements of <tt>R</tt>:<table class="examples"><tr><td><pre>i7 : a*b/b^4
a
o7 = --
3
b
o7 : F</pre>
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Fractions are reduced to the extent possible.<table class="examples"><tr><td><pre>i8 : f = (x-y)/(x^6-y^6)
1
o8 = ----------------------------------
5 4 3 2 2 3 4 5
x + x y + x y + x y + x*y + y
o8 : K</pre>
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<tr><td><pre>i9 : (x^3 - y^3) * f
x - y
o9 = -------
3 3
x + y
o9 : K</pre>
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The parts of a fraction may be extracted.<table class="examples"><tr><td><pre>i10 : numerator f
o10 = 1
o10 : R</pre>
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<tr><td><pre>i11 : denominator f
5 4 3 2 2 3 4 5
o11 = x + x y + x y + x y + x*y + y
o11 : R</pre>
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Alternatively, the functions <a href="_lift.html" title="lift to another ring">lift</a> and <a href="_liftable.html" title="whether lifting to another ring is possible">liftable</a> can be used.<table class="examples"><tr><td><pre>i12 : liftable(1/f,R)
o12 = true</pre>
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<tr><td><pre>i13 : liftable(f,R)
o13 = false</pre>
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<tr><td><pre>i14 : lift(1/f,R)
5 4 3 2 2 3 4 5
o14 = x + x y + x y + x y + x*y + y
o14 : R</pre>
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One can form resolutions and Gröbner bases of ideals in polynomial rings over fraction fields, as in the following example. Note that computations over fraction fields can be quite slow.<table class="examples"><tr><td><pre>i15 : S = K[u,v];</pre>
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<tr><td><pre>i16 : I = ideal(y^2*u^3 + x*v^3, u^2*v, u^4);
o16 : Ideal of S</pre>
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<tr><td><pre>i17 : gens gb I
o17 = | u2v u3+x/y2v3 v4 uv3 |
1 4
o17 : Matrix S <--- S</pre>
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<tr><td><pre>i18 : Ires = res I
1 3 2
o18 = S <-- S <-- S <-- 0
0 1 2 3
o18 : ChainComplex</pre>
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<tr><td><pre>i19 : Ires.dd_2
o19 = {3} | 0 y2/xuv |
{3} | -v2 -y2/xu2 |
{4} | u -v |
3 2
o19 : Matrix S <--- S</pre>
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One way to compute a blowup of an ideal <tt>I</tt> in <tt>R</tt>, is to compute the kernel of a map of a new polynomial ring into a fraction field of <tt>R</tt>, as shown below.<table class="examples"><tr><td><pre>i20 : A = ZZ/101[a,b,c];</pre>
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<tr><td><pre>i21 : f = map(K, A, {x^3/y^4, x^2/y^2, (x^2+y^2)/y^4});
o21 : RingMap K <--- A</pre>
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<tr><td><pre>i22 : kernel f
3 2 2 3 2 3 3
o22 = ideal (b c - a b - a , a*b c - a b*c - a c)
o22 : Ideal of A</pre>
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<div class="single"><h2>Caveat</h2>
<div>The input ring should be an integral domain.<p/>
Currently, for <tt>S</tt> as above, one cannot define <tt>frac S</tt> or fractions <tt>u/v</tt>. One can get around that by defining <tt>B = ZZ/101[x,y,u,v]</tt> and identify <tt>frac S</tt> with <tt>frac B</tt>.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_numerator.html" title="numerator of a fraction">numerator</a> -- numerator of a fraction</span></li>
<li><span><a href="_denominator.html" title="denominator of a fraction">denominator</a> -- denominator of a fraction</span></li>
<li><span><a href="_liftable.html" title="whether lifting to another ring is possible">liftable</a> -- whether lifting to another ring is possible</span></li>
<li><span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>frac</tt> :</h2>
<ul><li>frac(EngineRing)</li>
<li>frac(FractionField)</li>
<li>frac(Ring)</li>
</ul>
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