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<head><title>Applications to multigraded polynomial rings -- finding Heft</title>
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<div><h1>Applications to multigraded polynomial rings -- finding Heft</h1>
<div>A vector configuration is <em>acyclic</em> if it has a positive linear functional. Using <tt>fourierMotzkin</tt> we can determine if a vector configuration has a positive linear functional.<p/>
Given an acyclic vector configuration (as a list of lists of <a href="../../Macaulay2Doc/html/___Z__Z.html" title="the class of all integers">ZZ</a> or <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a>), the function <tt>findHeft</tt> finds a <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> representing a positive linear functional.<table class="examples"><tr><td><pre>i1 : findHeft = vectorConfig -> (
A := transpose matrix vectorConfig;
B := (fourierMotzkin A)#0;
r := rank source B;
heft := first entries (matrix{toList(r:-1)} * transpose B);
g := gcd heft;
if g > 1 then heft = apply(heft, h -> h //g);
heft);</pre>
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If a polynomial ring as a multigrading determined by a vector configuration, then a positive linear functional plays the role of a <a href="../../Macaulay2Doc/html/___Heft.html" title="name for an optional argument">Heft</a> vector.<p/>
For example, <tt>S</tt> is the Cox homogeneous coordinate ring for second Hirzebruch surface (under an appropriate choice of basis for the Picard group).<table class="examples"><tr><td><pre>i2 : vectorConfig = {{1,0},{-2,1},{1,0},{0,1}}
o2 = {{1, 0}, {-2, 1}, {1, 0}, {0, 1}}
o2 : List</pre>
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<tr><td><pre>i3 : hft = findHeft vectorConfig
o3 = {1, 3}
o3 : List</pre>
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<tr><td><pre>i4 : S = QQ[x_1,x_2,y_1,y_2, Heft => hft, Degrees => vectorConfig];</pre>
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<tr><td><pre>i5 : irrelevantIdeal = intersect(ideal(x_1,x_2), ideal(y_1,y_2))
o5 = ideal (x y , x y , x y , x y )
2 1 1 1 2 2 1 2
o5 : Ideal of S</pre>
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<tr><td><pre>i6 : res (S^1/irrelevantIdeal)
1 4 4 1
o6 = S <-- S <-- S <-- S <-- 0
0 1 2 3 4
o6 : ChainComplex</pre>
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The Betti numbers correspond to the f-vector of the polytope associated to the second Hirzebruch surface.<p/>
Similarly, <tt>R</tt> is the Cox homogeneous coordinate ring for the blowup of the projective plane at three points (under an appropriate choice of basis for the Picard group).<table class="examples"><tr><td><pre>i7 : vectorConfig = {{1,0,0,0},{0,1,0,0},{0,-1,1,0},{0,1,-1,1},
{1,0,-1,1},{-1,0,0,1}}
o7 = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, -1, 1, 0}, {0, 1, -1, 1}, {1, 0, -1,
------------------------------------------------------------------------
1}, {-1, 0, 0, 1}}
o7 : List</pre>
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<tr><td><pre>i8 : hft = findHeft vectorConfig
o8 = {4, 4, 7, 7}
o8 : List</pre>
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<tr><td><pre>i9 : R = QQ[x_1..x_6, Heft => hft, Degrees => vectorConfig];</pre>
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<tr><td><pre>i10 : irrelevantIdeal = ideal(x_3*x_4*x_5*x_6,x_1*x_4*x_5*x_6,x_1*x_2*x_5*x_6,
x_1*x_2*x_3*x_6,x_2*x_3*x_4*x_5,x_1*x_2*x_3*x_4)
o10 = ideal (x x x x , x x x x , x x x x , x x x x , x x x x , x x x x )
3 4 5 6 1 4 5 6 1 2 5 6 1 2 3 6 2 3 4 5 1 2 3 4
o10 : Ideal of R</pre>
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<tr><td><pre>i11 : res (R^1/irrelevantIdeal)
1 6 6 1
o11 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o11 : ChainComplex</pre>
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Again, the Betti numbers correspond to the f-vector of the polytope associated to this toric variety.<p/>
For more information about resolutions of the irrelevant ideal of a toric variety, see subsection 4.3.6 in <a href="http://www.math.umn.edu/~ezra/">Ezra Miller</a> and <a href="http://math.berkeley.edu/~bernd/">Bernd Sturmfels'</a> <em>Combinatorial commutative algebra</em>, Graduate Texts in Mathematics 277, Springer-Verlag, New York, 2005.<p/>
For more information about vector configurations is subsections 6.2 & 6.4 in <a href="http://www.math.tu-berlin.de/~ziegler/">Gunter M. Ziegler's</a> <em>Lectures on Polytopes</em>, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995.</div>
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