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<head><title>degreesRing(List) -- the ring of degrees</title>
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<div><h1>degreesRing(List) -- the ring of degrees</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>degreesRing x</tt></div>
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<li><span>Function: <a href="_degrees__Ring.html" title="the ring of degrees">degreesRing</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>x</tt>, <span>a <a href="___List.html">list</a></span>, <span>a <a href="___List.html">list</a></span> or <span>an <a href="___Z__Z.html">integer</a></span>a list of integers, or a single integer</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Polynomial__Ring.html">polynomial ring</a></span>, the ring of Laurent polynomials whose monomials correspond to the degrees of the monomials in a ring whose heft vector is <tt>x</tt>, or, if <tt>x</tt> is an integer, in a ring with degree rank <tt>x</tt> and no heft vector. See <tt>degreeMonoid</tt> (missing documentation<!-- tag: degreeMonoid -->).</span></li>
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<div class="single"><h2>Description</h2>
<div><p>This function produces a Laurent polynomial ring in <tt>n</tt> variables<tt>T_0, ... , T_{n-1}</tt>, where <tt>n</tt> is the length of <tt>x</tt> if <tt>x</tt> is a list and is <tt>x</tt> otherwise. If <tt>n=1</tt>, then the single variable is <tt>T</tt>. Use <a href="_use.html" title="install or activate object">use</a> as in the following example to assign the indeterminates of the ring to global variables, or assign the ring itself to a global variable.</p>
<table class="examples"><tr><td><pre>i1 : degreesRing 3
o1 = ZZ[T , T , T ]
0 1 2
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : describe oo
o2 = ZZ[T , T , T , Degrees => {3:{}}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 0, Inverses => true, Global => false]
0 1 2 {Weights => {3:-1} }
{GroupLex => 3 }
{Position => Up }</pre>
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<tr><td><pre>i3 : T_0
o3 = T
0
o3 : IndexedVariable</pre>
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<tr><td><pre>i4 : use degreesRing 3
o4 = ZZ[T , T , T ]
0 1 2
o4 : PolynomialRing</pre>
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<tr><td><pre>i5 : T_0
o5 = T
0
o5 : ZZ[T , T , T ]
0 1 2</pre>
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<p>Elements of this ring are used as variables for Poincare polynomials generated by <a href="_poincare.html" title="assemble degrees into polynomial">poincare</a> and <a href="_poincare__N.html" title="assemble degrees into polynomial">poincareN</a> as well as <a href="_hilbert__Series.html">Hilbertseries</a>.</p>
<p>The degrees ring is a Laurent polynomial ring, as can be seen by the option in the definition of the ring that says <tt>Inverses => true</tt>. The monomial ordering used in the degrees ring is <tt>RevLex</tt> so the polynomials in it will be displayed with the smallest exponents first, because such polynomials are often used as Hilbert series.</p>
<table class="examples"><tr><td><pre>i6 : W = degreesRing {1,2,5}
o6 = W
o6 : PolynomialRing</pre>
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<tr><td><pre>i7 : describe W
o7 = ZZ[T , T , T , Degrees => {1..2, 5}, MonomialOrder => {MonomialSize => 32 }, DegreeRank => 1, Inverses => true, Global => false]
0 1 2 {Weights => {-1, -2, -5}}
{GroupLex => 3 }
{Position => Up }</pre>
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<tr><td><pre>i8 : use W
o8 = W
o8 : PolynomialRing</pre>
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<tr><td><pre>i9 : (1+T_1+T_2^2)^3
2 3 2 2 2 2 4 4 6
o9 = 1 + 3T + 3T + T + 3T + 6T T + 3T T + 3T + 3T T + T
1 1 1 2 1 2 1 2 2 1 2 2
o9 : W</pre>
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<tr><td><pre>i10 : degreesRing 3
o10 = ZZ[T , T , T ]
0 1 2
o10 : PolynomialRing</pre>
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<tr><td><pre>i11 : describe oo
o11 = ZZ[T , T , T , Degrees => {3:{}}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 0, Inverses => true, Global => false]
0 1 2 {Weights => {3:-1} }
{GroupLex => 3 }
{Position => Up }</pre>
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<tr><td><pre>i12 : R = QQ[x,y,Degrees => {{1,-2},{2,-1}}];</pre>
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<tr><td><pre>i13 : heft R
o13 = {1, 0}
o13 : List</pre>
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<tr><td><pre>i14 : describe degreesRing R
o14 = ZZ[T , T , Degrees => {1, 0}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1, Inverses => true, Global => false]
0 1 {Weights => {-1..0}}
{GroupLex => 2 }
{Position => Up }</pre>
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<tr><td><pre>i15 : S = QQ[x,y,Degrees => {-2,1}];</pre>
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<tr><td><pre>i16 : heft S</pre>
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<tr><td><pre>i17 : describe degreesRing S
o17 = ZZ[T, Degrees => {{}}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 0, Inverses => true, Global => false]
{Weights => {-1} }
{GroupLex => 1 }
{Position => Up }</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_poincare.html" title="assemble degrees into polynomial">poincare</a> -- assemble degrees into polynomial</span></li>
<li><span><a href="_poincare__N.html" title="assemble degrees into polynomial">poincareN</a> -- assemble degrees into polynomial</span></li>
<li><span><a href="_hilbert__Function.html" title="the Hilbert function">hilbertFunction</a> -- the Hilbert function</span></li>
<li><span><a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</a> -- compute the Hilbert series</span></li>
<li><span><a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a> -- compute the Hilbert polynomial</span></li>
<li><span><a href="_reduce__Hilbert.html" title="reduce a Hilbert series expression">reduceHilbert</a> -- reduce a Hilbert series expression</span></li>
<li><span><a href="_division_spin_sppolynomial_springs_spwith_spmonomials_spless_spthan_sp1.html" title="">division in polynomial rings with monomials less than 1</a></span></li>
<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="_use.html" title="install or activate object">use</a> -- install or activate object</span></li>
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