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<head><title>LPP -- return the lex-plus-powers (LPP) ideal corresponding to a given Hilbert function and power sequence</title>
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<div><h1>LPP -- return the lex-plus-powers (LPP) ideal corresponding to a given Hilbert function and power sequence</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>L=LPP(R,hilb,power)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span></span></li>
<li><span><tt>hilb</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a Hilbert function as a list</span></li>
<li><span><tt>power</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of positive integers in weakly increasing order</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>L</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, an LPP ideal with the desired Hilbert function and power sequence if one exists and <tt>null</tt> otherwise</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Let <tt>a<sub>1</sub> <= ... <= a<sub>n</sub></tt> be positive integers. A monomial ideal <tt>L</tt> in a polynomial ring <tt>R=k[x<sub>1</sub>,...,x<sub>n</sub>]</tt> is called an <tt>(a<sub>1</sub>,...a<sub>n</sub>)</tt>-lex-plus-powers (LPP) ideal if it satisfies two conditions:</p>
<p>(1) L is minimally generated by <tt>x<sub>1</sub><sup>a<sub>1</sub></sup>, ..., x<sub>n</sub><sup>a<sub>n</sub></sup></tt> and monomials <tt>m<sub>1</sub>, ..., m<sub>t</sub></tt>. (2) Suppose that r is a monomial such that <tt>deg r = deg m<sub>i</sub></tt>, and <tt>r > m<sub>i</sub></tt> in the lex order. Then <tt>r</tt> is in <tt>L</tt>.</p>
<p>LPP ideals are generalizations of Artinian lexicographic ideals. Condition (2) represents the lex portion of the LPP ideal; monomials that are not powers of a variable must satisfy a lexicographic condition similar to what generators of lex ideals satisfy. An LPP ideal also contains powers of all the variables in weakly increasing order.</p>
<div>LPP ideals arise in conjectures of Eisenbud-Green-Harris and Charalambous-Evans in algebraic geometry, Hilbert functions, and graded free resolutions. They are conjectured to play a role analogous to that of lexicographic ideals in theorems of Macaulay on Hilbert functions and Bigatti, Hulett, and Pardue on resolutions.</div>
<table class="examples"><tr><td><pre>i1 : R=ZZ/32003[a..c];</pre>
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<tr><td><pre>i2 : LPP(R,{1,3,6,5,3},{3,3,4})
3 3 4 2 2 2 2 2 3
o2 = ideal (a , b , c , a b, a c, a*b , a*b*c , b c )
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : LPP(R,{1,3,4,2,1},{2,3,5}) --an Artinian lex ideal
2 3 5 2 2 3
o3 = ideal (a , b , c , a*b, a*c , b c, b*c )
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : LPP(R,{1,3,4,2,1},{2,4,3}) --exponents not in weakly increasing order</pre>
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<tr><td><pre>i5 : LPP(R,{1,3,4,2,1},{2,2,3}) --no LPP ideal with this Hilbert function and power sequence</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_generate__L__P__Ps.html" title="return all LPP ideals corresponding to a given Hilbert function">generateLPPs</a> -- return all LPP ideals corresponding to a given Hilbert function</span></li>
<li><span><a href="_is__L__P__P.html" title="determine whether an ideal is an LPP ideal">isLPP</a> -- determine whether an ideal is an LPP ideal</span></li>
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<div class="waystouse"><h2>Ways to use <tt>LPP</tt> :</h2>
<ul><li>LPP(PolynomialRing,List,List)</li>
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