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<head><title>Weights -- assigning weights to the variables</title>
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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_monomial_sporderings.html" title="">monomial orderings</a> > <a href="___Weights.html" title="assigning weights to the variables">Weights</a></div>
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<div><h1>Weights -- assigning weights to the variables</h1>
<div class="single"><h2>Description</h2>
<div>Given a list L of n integers, the weight order on a polynomial ring in n variables is defined by: x^A > x^B if A_1 L_1 + ... + A_n L_n > B_1 L_1 + ... + B_n L_n.<p/>
The leading component of a polynomial under a weight order need not be a monomial. When two monomials have the same weight, by default they are further distinguished with the GRevLex order.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d,MonomialOrder=>{Weights => {-1,2,3,4}},Global=>false];</pre>
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<tr><td><pre>i2 : f = a^2 + b+ c^2 + b*d
2 2
o2 = c + b*d + b + a
o2 : R</pre>
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<tr><td><pre>i3 : leadTerm f
2
o3 = c
o3 : R</pre>
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However, we can retrieve the entire leading component with the command <tt>leadTerm(1,f)</tt>. The plain <tt>leadTerm f</tt> is in this case the same as <tt>leadTerm(2,f)</tt>: they both use the full specification of the monomial ordering, first by weight and then by the <tt>GRevLex</tt> order. In contrast, <tt>leadTerm(1,f)</tt>only distinguishes monomials by the first, i.e., weight, specification.<table class="examples"><tr><td><pre>i4 : leadTerm(1,ideal(f))
o4 = | c2+bd |
1 1
o4 : Matrix R <--- R</pre>
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The weight order may be combined with further ordering elements to break ties. In the following example, we use a second weight vector to break ties under first weight vector.<table class="examples"><tr><td><pre>i5 : R = QQ[a..d,MonomialOrder=>{Weights => {1,2,3,4}, Weights => {2,4,2,1}}];</pre>
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<tr><td><pre>i6 : f = a^6 + b^3+ c^2
6 3 2
o6 = a + b + c
o6 : R</pre>
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<tr><td><pre>i7 : leadTerm(f)
6
o7 = a
o7 : R</pre>
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<tr><td><pre>i8 : leadTerm(1, ideal(f))
o8 = | a6+b3+c2 |
1 1
o8 : Matrix R <--- R</pre>
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<tr><td><pre>i9 : leadTerm(2, ideal(f))
o9 = | a6+b3 |
1 1
o9 : Matrix R <--- R</pre>
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<tr><td><pre>i10 : leadTerm(3, ideal(f))
o10 = | a6 |
1 1
o10 : Matrix R <--- R</pre>
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<tt>leadTerm(3, ideal(f))</tt> uses both the specified weights and the <tt>GRevLex</tt> order to calculate the leading component.<p/>
When the number of weights is smaller than the number of variables, the remaining variables are given weight 0. In the following example, <tt>c</tt> and <tt>d</tt> have weight 0.<table class="examples"><tr><td><pre>i11 : R = QQ[a..d,MonomialOrder=>{Weights => {1,2}, Lex}];</pre>
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<tr><td><pre>i12 : f = a^2 + b+ c^2 + b*d
2 2
o12 = a + b*d + b + c
o12 : R</pre>
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<div class="single"><h2>Functions with optional argument named Weights :</h2>
<ul><li><span><a href="_betti_lp..._cm_sp__Weights_sp_eq_gt_sp..._rp.html" title="">betti(..., Weights => ...)</a></span></li>
<li><span>monoid(..., Weights => ...), see <span><a href="_monoid.html" title="make or retrieve a monoid">monoid</a> -- make or retrieve a monoid</span></span></li>
<li><span>newRing(..., Weights => ...), see <span><a href="_new__Ring.html" title="make a copy of a ring, with some features changed">newRing</a> -- make a copy of a ring, with some features changed</span></span></li>
<li><span><a href="_regularity_lp..._cm_sp__Weights_sp_eq_gt_sp..._rp.html" title="">regularity(..., Weights => ...)</a></span></li>
<li><span>symmetricAlgebra(..., Weights => ...), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span>tensor(..., Weights => ...), see <span><a href="_tensor_lp__Ring_cm__Ring_rp.html" title="tensor product">tensor(Ring,Ring)</a> -- tensor product</span></span></li>
</ul>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_lead__Term.html" title="get the greatest term">leadTerm</a> -- get the greatest term</span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Weights.html" title="assigning weights to the variables">Weights</a> is <span>a <a href="___Symbol.html">symbol</a></span>.</p>
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