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<head><title>Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis</title>
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<div><h1>Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis</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>f % G</tt></div>
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<li><span>Operator: <a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a></span></li>
<li><div class="single">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>G</tt>, <span>a <a href="___Groebner__Basis.html">Groebner basis</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span>the normal form of <tt>f</tt> with respect to the partially computed Gröbner basis <tt>G</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>In the following example, the seventh power of the trace of the matrix M is in the ideal generated by the entries of the cube of M. Since the ideal I is homogeneous, it is only required to compute the Gröbner basis in degrees at most seven.<table class="examples"><tr><td><pre>i1 : R = QQ[a..i];</pre>
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<tr><td><pre>i2 : M = genericMatrix(R,a,3,3)
o2 = | a d g |
| b e h |
| c f i |
3 3
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : I = ideal(M^3);
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : f = trace M
o4 = a + e + i
o4 : R</pre>
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<tr><td><pre>i5 : G = gb(I, DegreeLimit=>3)
o5 = GroebnerBasis[status: DegreeLimit; all S-pairs handled up to degree 3]
o5 : GroebnerBasis</pre>
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<tr><td><pre>i6 : f^7 % G == 0
o6 = false</pre>
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<tr><td><pre>i7 : gb(I, DegreeLimit=>7)
o7 = GroebnerBasis[status: DegreeLimit; all S-pairs handled up to degree 7]
o7 : GroebnerBasis</pre>
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<tr><td><pre>i8 : f^7 % G
o8 = 0
o8 : R</pre>
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<tr><td><pre>i9 : gb I
o9 = GroebnerBasis[status: done; S-pairs encountered up to degree 9]
o9 : GroebnerBasis</pre>
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In these homogeneous situations, Macaulay2 only computes the Gröbner basis as far as required, as shown below.<table class="examples"><tr><td><pre>i10 : I = ideal(M^3);
o10 : Ideal of R</pre>
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<tr><td><pre>i11 : G = gb(I, StopBeforeComputation=>true)
o11 = GroebnerBasis[status: not started; all S-pairs handled up to degree -1]
o11 : GroebnerBasis</pre>
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<tr><td><pre>i12 : f^7 % I
o12 = 0
o12 : R</pre>
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<tr><td><pre>i13 : status G
o13 = status: DegreeLimit; all S-pairs handled up to degree 7</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></li>
<li><span><a href="___Gröbner_spbases.html" title="">Gröbner bases</a></span></li>
<li><span><a href="_generic__Matrix.html" title="make a generic matrix of variables">genericMatrix</a> -- make a generic matrix of variables</span></li>
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