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<head><title>coefficients -- monomials and their coefficients</title>
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<div><h1>coefficients -- monomials and their coefficients</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>(monoms,coeffs) = coefficients(f,Variables=>v,Monomials=>m)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, either a one row <a href="___Matrix.html" title="the class of all matrices">Matrix</a> or a <a href="___Ring__Element.html" title="the class of all ring elements handled by the engine">RingElement</a></span></li>
<li><span><tt>v</tt>, <span>a <a href="___List.html">list</a></span>, of variables</span></li>
<li><span><tt>m</tt>, either a list of monomials, or a one row matrix of monomials</span></li>
</ul>
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<li><div class="single">Outputs:<ul><li><span><tt>monoms</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a one row matrix of the monomials appearing in <tt>f</tt></span></li>
<li><span><tt>coeffs</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a matrix with the coefficients of <tt>monoms</tt> appearing in <tt>f</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_coefficients_lp..._cm_sp__Monomials_sp_eq_gt_sp..._rp.html">Monomials => ...</a>, -- specify monomials</span></li>
<li><span><a href="_coefficients_lp..._cm_sp__Variables_sp_eq_gt_sp..._rp.html">Variables => ...</a>, -- take coefficients using these variables</span></li>
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<div class="single"><h2>Description</h2>
<div>If the optional argument <tt>Variables=>v</tt> is given, then the monomials will only involve these variables, and the coefficients will involve only the other variables.<p/>
If the optional argument <tt>Monomials=>m</tt> is not given, then the set of monomials appearing in <tt>f</tt> is calculated using <a href="_monomials.html" title="matrix of monomials in a ring element or matrix">monomials</a>.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,a,b,c,d,e,f];</pre>
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<tr><td><pre>i2 : F = a*x^2+b*x*y+c*y^2
2 2
o2 = x a + x*y*b + y c
o2 : R</pre>
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<tr><td><pre>i3 : (M,C) = coefficients(F, Variables=>{x,y})
o3 = (| x2 xy y2 |, {2} | a |)
{2} | b |
{2} | c |
o3 : Sequence</pre>
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The resulting matrices have the following property.<table class="examples"><tr><td><pre>i4 : M*C == matrix{{F}}
o4 = true</pre>
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<p/>
The Sylvester matrix of two generic quadratic forms.<table class="examples"><tr><td><pre>i5 : G = d*x^2+e*x*y+f*y^2
2 2
o5 = x d + x*y*e + y f
o5 : R</pre>
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<tr><td><pre>i6 : P = matrix{{x*F,y*F,x*G,y*G}}
o6 = | x3a+x2yb+xy2c x2ya+xy2b+y3c x3d+x2ye+xy2f x2yd+xy2e+y3f |
1 4
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : (M,C) = coefficients(P, Variables=>{x,y})
o7 = (| x3 x2y xy2 y3 |, {3} | a 0 d 0 |)
{3} | b a e d |
{3} | c b f e |
{3} | 0 c 0 f |
o7 : Sequence</pre>
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<tr><td><pre>i8 : M*C == P
o8 = true</pre>
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We may give the monomials directly. This is useful if we are taking coefficients of several elements or matrices, and need a consistent choice of monomials.<table class="examples"><tr><td><pre>i9 : (M,C) = coefficients(P, Variables=>{x,y}, Monomials=>{x^3,y^3,x^2*y,x*y^2})
o9 = (| x3 y3 x2y xy2 |, {3} | a 0 d 0 |)
{3} | 0 c 0 f |
{3} | b a e d |
{3} | c b f e |
o9 : Sequence</pre>
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If not all of the monomials are used, then <tt>M*C == P</tt> no longer holds.<table class="examples"><tr><td><pre>i10 : (M,C) = coefficients(P, Variables=>{x,y}, Monomials=>{x^3,y^3})
o10 = (| x3 y3 |, {3} | a 0 d 0 |)
{3} | 0 c 0 f |
o10 : Sequence</pre>
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<tr><td><pre>i11 : M*C == P
o11 = false</pre>
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<div class="single"><h2>Caveat</h2>
<div>Currently, the matrix <tt>f</tt> must have only one row. This restriction will hopefully be lifted in the future.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_monomials.html" title="matrix of monomials in a ring element or matrix">monomials</a> -- matrix of monomials in a ring element or matrix</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>coefficients</tt> :</h2>
<ul><li>coefficients(Matrix)</li>
<li>coefficients(RingElement)</li>
</ul>
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