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<div><h1>heft vectors</h1>
<div><p>A <em>heft vector</em> for a polynomial ring is a vector with integer entries, of the same length as the degree vectors of the variables of the ring, whose dot product with each of them is (strictly) positive. Unless one is specified explicitly, then a good one will be found automatically. The heft vector is used in various internal algorithms, such as the one in <a href="_basis.html" title="basis of all or part of a module or ring">basis</a>, as a way of organizing the sequence of steps, proceeding incrementally to larger values of the dot product of the degree of a monomial with the heft vector.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : degrees R
o2 = {{1}, {1}, {1}, {1}}
o2 : List</pre>
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<tr><td><pre>i3 : heft R
o3 = {1}
o3 : List</pre>
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<tr><td><pre>i4 : S = QQ[a..d,DegreeRank => 4];</pre>
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<tr><td><pre>i5 : degrees S
o5 = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}
o5 : List</pre>
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<tr><td><pre>i6 : heft S
o6 = {1, 1, 1, 1}
o6 : List</pre>
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<tr><td><pre>i7 : T = QQ[a,b,Degrees => {1,-1}]
o7 = T
o7 : PolynomialRing</pre>
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<tr><td><pre>i8 : degrees T
o8 = {{1}, {-1}}
o8 : List</pre>
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<tr><td><pre>i9 : heft T</pre>
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<tr><td><pre>i10 : U = QQ[a..d,Degrees => {{2,0},{1,-1},{0,-2},{-1,-3}}]
o10 = U
o10 : PolynomialRing</pre>
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<tr><td><pre>i11 : degrees U
o11 = {{2, 0}, {1, -1}, {0, -2}, {-1, -3}}
o11 : List</pre>
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<tr><td><pre>i12 : heft U
o12 = {3, -2}
o12 : List</pre>
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<p>The heft vector, multiplied by -1, is used as the weight vector in the monomial ordering of the degrees ring, and the <em>order</em> of the series expansions of the Hilbert series refers to the weight formed with respect to that weight vector.</p>
<table class="examples"><tr><td><pre>i13 : hilbertSeries U
1
o13 = ----------------------------------------
2 -1 -2 -1 -3
(1 - T )(1 - T T )(1 - T )(1 - T T )
0 0 1 1 0 1
o13 : Expression of class Divide</pre>
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<tr><td><pre>i14 : describe ring numerator oo
o14 = ZZ[T , T , Degrees => {3, -2}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1, Inverses => true, Global => false]
0 1 {Weights => {-3, 2}}
{GroupLex => 2 }
{Position => Up }</pre>
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<tr><td><pre>i15 : hilbertSeries(U,Order => 8)
-1 -3 -2 -1 2 -2 -6 -1 -5
o15 = 1 + T T + T + T T + T + T T + T T
0 1 1 0 1 0 0 1 0 1
o15 : ZZ[T , T ]
0 1</pre>
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<p>The heft vector is used in the computation of degrees of modules over a polynomial ring <tt>R</tt>, because it gives a homomorphism from the degrees ring of <tt>R</tt> to the Laurent polynomial ring in one variable <tt>T</tt> that sends monomials corresponding to the degrees of variables of <tt>R</tt> to positive powers of <tt>T</tt>. The degree is then defined to be the coefficient of the lead term in the expansion of the Hilbert series in powers of <tt>1-T</tt>.</p>
<table class="examples"><tr><td><pre>i16 : R = QQ[x,y,Heft=>{3}];</pre>
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<tr><td><pre>i17 : degree R
1
o17 = -
9
o17 : QQ</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_heft.html" title="heft vector of ring, module, graded module, or resolution">heft</a> -- heft vector of ring, module, graded module, or resolution</span></li>
<li><span><a href="_monoid.html" title="make or retrieve a monoid">monoid(..., Heft => ...)</a> -- make or retrieve a monoid</span></li>
<li><span><a href="_degrees__Ring.html" title="the ring of degrees">degreesRing</a> -- the ring of degrees</span></li>
<li><span><a href="_multidegree.html" title="multidegree">multidegree</a> -- multidegree</span></li>
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