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<div><h1>methods for normal forms and remainder -- calculate the normal form of ring elements and matrices</h1>
<div><div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>f % I</tt></div>
</dd></dl>
</div>
</li>
<li>Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Ring__Element.html">ring element</a></span>, or <span>a <a href="___Matrix.html">matrix</a></span></span></li>
<li><span><tt>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, <span>a <a href="___Matrix.html">matrix</a></span>, or <span>a <a href="___Ring__Element.html">ring element</a></span></span></li>
</ul>
</li>
<li>Outputs:<ul><li><span>the normal form of <tt>f</tt> with respect to a Gröbner basis of I</span></li>
</ul>
</li>
</ul>
</div>
The result has the same type as <tt>f</tt>. The normal form of a matrix is a matrix of the same shape whose columns have been reduced to normal form by the Gröbner basis of <tt>I</tt>.<p/>
To reduce <tt>f</tt> with respect to <tt>I</tt>, a (partial) Gröbner basis of <tt>I</tt> is computed, unless it has already been done, or unless <tt>I</tt> is a <a href="___Monomial__Ideal.html" title="the class of all monomial ideals handled by the engine">MonomialIdeal</a>.<table class="examples"><tr><td><pre>i1 : R = ZZ/1277[x,y];</pre>
</td></tr>
<tr><td><pre>i2 : I = ideal(x^3 - 2*x*y, x^2*y - 2*y^2 + x);
o2 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i3 : (x^3 - 2*x) % I
o3 = -2x
o3 : R</pre>
</td></tr>
<tr><td><pre>i4 : (x^3) % I
o4 = 0
o4 : R</pre>
</td></tr>
<tr><td><pre>i5 : S = ZZ[x,y];</pre>
</td></tr>
<tr><td><pre>i6 : 144*x^2*y^2 % (7*x*y-2)
2 2
o6 = - 3x y + 12
o6 : S</pre>
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<p/>
If <tt>I</tt> is a matrix between free modules, then a Gröbner basis of <tt>I</tt> is a Gröbner basis of the submodule generated by the columns of the matrix.<table class="examples"><tr><td><pre>i7 : S = QQ[a..f]
o7 = S
o7 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i8 : J = ideal(a*b*c-d*e*f,a*b*d-c*e*f, a*c*e-b*d*f)
o8 = ideal (a*b*c - d*e*f, a*b*d - c*e*f, a*c*e - b*d*f)
o8 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i9 : C = res J
1 3 6 6 2
o9 = S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5
o9 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i10 : F = syz transpose C.dd_4
o10 = {-8} | 0 0 a de ce bd bc 0 0 0 0 |
{-8} | 0 b 0 df cf 0 0 ad ac 0 0 |
{-8} | c 0 0 -ef 0 -bf 0 -ae 0 ab 0 |
{-8} | d 0 0 0 ef 0 bf 0 ae 0 ab |
{-8} | 0 e 0 0 0 -df -cf 0 0 ad -ac |
{-8} | 0 0 f 0 0 0 0 -de -ce bd -bc |
6 11
o10 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i11 : G = transpose C.dd_3
o11 = {-8} | bd -ce de a 0 0 |
{-8} | -ac -cf df 0 -b 0 |
{-8} | -bf -ab -ef 0 0 -c |
{-8} | -ae -ef -ab 0 0 -d |
{-8} | -df -ad ac 0 -e 0 |
{-8} | ce -bd bc f 0 0 |
6 6
o11 : Matrix S <--- S</pre>
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Since <tt>C</tt> is a complex, we know that the image of <tt>G</tt> is contained in the image of <tt>F</tt>.<table class="examples"><tr><td><pre>i12 : G % F
o12 = 0
6 6
o12 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i13 : F % G
o13 = {-8} | 0 0 0 de ce bd bc 0 bd -ce de |
{-8} | 0 0 0 df cf 0 0 ad 0 -cf df |
{-8} | 0 0 0 -ef 0 -bf 0 -ae -bf 0 -ef |
{-8} | 0 0 0 0 ef 0 bf 0 0 -ef 0 |
{-8} | 0 0 0 0 0 -df -cf 0 -df 0 0 |
{-8} | 0 0 0 0 0 0 0 -de 0 0 0 |
6 11
o13 : Matrix S <--- S</pre>
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The inclusion is strict since <tt>F % G != 0</tt> shows that the image of <tt>F</tt> is not contained in the image of <tt>G</tt>.<p/>
Normal forms work over quotient rings too.<table class="examples"><tr><td><pre>i14 : A = QQ[x,y,z]/(x^3-y^2-3)
o14 = A
o14 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i15 : I = ideal(x^4, y^4)
2 4
o15 = ideal (x*y + 3x, y )
o15 : Ideal of A</pre>
</td></tr>
<tr><td><pre>i16 : J = ideal(x^3*y^2, x^2*y^3)
4 2 2 3
o16 = ideal (y + 3y , x y )
o16 : Ideal of A</pre>
</td></tr>
<tr><td><pre>i17 : (gens J) % I
o17 = 0
1 2
o17 : Matrix A <--- A</pre>
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<p/>
Here is an example involving rational functions.<table class="examples"><tr><td><pre>i18 : kk = frac(ZZ[a,b])
o18 = kk
o18 : FractionField</pre>
</td></tr>
<tr><td><pre>i19 : B = kk[x,y,z]
o19 = B
o19 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i20 : I = ideal(a*x^2-b*x-y-1, 1/b*y^2-z-1)
2 1 2
o20 = ideal (a*x - b*x - y - 1, -y - z - 1)
b
o20 : Ideal of B</pre>
</td></tr>
<tr><td><pre>i21 : gens gb I
o21 = | y2-bz-b x2-b/ax-1/ay-1/a |
1 2
o21 : Matrix B <--- B</pre>
</td></tr>
<tr><td><pre>i22 : x^2*y^2 % I
2 2
b b b b b b
o22 = --x*z + -y*z + --x + -y + -z + -
a a a a a a
o22 : B</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> -- a binary operator, usually used for remainder and reduction</span></li>
<li><span><a href="___Gröbner_spbases.html" title="">Gröbner bases</a></span></li>
<li><span><a href="_generators.html" title="provide matrix or list of generators">generators</a> -- provide matrix or list of generators</span></li>
<li><span><a href="___Matrix_sp_pc_sp__Groebner__Basis.html" title="calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis">Matrix % GroebnerBasis</a> -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis</span></li>
</ul>
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