Difference between revisions of "ApCoCoA-1:NC.HF"

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{{Version|1}}
 
<command>
 
<command>
 
<title>NC.HF</title>
 
<title>NC.HF</title>
 
<short_description>
 
<short_description>
Hilbert function of <tt>K</tt>-algebra.
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Enumerate the values of the Hilbert-Dehn function of a finitely generated <tt>K</tt>-algebra.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
NC.HF(Gb:LIST):LIST
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NC.HF(G:LIST[, DB:INT]):LIST
NC.HF(Gb:LIST, DegreeBound:INT):LIST
 
 
</syntax>
 
</syntax>
 
<description>
 
<description>
 +
Let <tt>P</tt> be a finitely generated non-commutative polynomial ring over <tt>K</tt>, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>P</tt>. Then <tt>P/I</tt> is a finitely generated <tt>K</tt>-algebra. Moreover, for every integer <tt>i</tt>, we let <tt>F_{i}</tt> be the <tt>K</tt>-vector subspace generated by the words of length less than or equal to <tt>i</tt>. Clearly, the set <tt>{F_{i}}</tt> is a filtration of <tt>P</tt>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>P/I</tt>. The <em>Hilbert-Dehn function</em> of <tt>K</tt>-algebra <tt>P/I</tt> is a map <tt>HF: N --&gt; N</tt> defined by <tt>HF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I))</tt>, i.e. <tt>HF(i)</tt> is equal to the number of words of length <tt>i</tt> in a Macaulay's basis (see <ref>ApCoCoA-1:NC.MB|NC.MB</ref>) of <tt>P/I</tt>.
 +
<par/>
 
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
<par/>
 
<par/>
Please set ring environment coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and ordering through the functions <ref>NC.SetFp</ref>(Prime), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(Ordering), respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
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Please set non-commutative polynomial ring (via the command <ref>ApCoCoA-1:Use|Use</ref>) and word ordering (via the function <ref>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.
 +
<itemize>
 +
<item>@param <em>G:</em> a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST []. <em>Warning:</em> users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!</item>
 +
<item>@return: a LIST of non-negative integers, which are values of the Hilbert-Dehn function of the K-algebra <tt>P/&lt;G&gt;</tt>.</item>
 +
</itemize>
 +
Optional parameter:
 
<itemize>
 
<itemize>
<item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> which is a Groebner basis of (two-sided) ideal generated by Gb. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Then, polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST [].</item>
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<item>@param <em>DB:</em> a positive INT, which is a degree bound of the Hilbert-Dehn function. <em>Note that</em> we set DB=32 by default. Thus, in the case that the <tt>K</tt>-dimension of <tt>P/&lt;G&gt;</tt> is finite, it is necessary to set DB to a large enough INT in order to compute all the values of the Hilbert-Dehn function.</item>
<item>@param <em>DegreeBound:</em> (optional) a positive integer which is a degree bound of Hilbert funtion.</item>
 
<item>@return: a LIST of non-negative integers, which is a list of coefficient of Hilbert function of <tt>K&lt;X&gt;/(Gb)</tt>.</item>
 
 
</itemize>
 
</itemize>
 
<example>
 
<example>
NC.SetX(<quotes>abc</quotes>);
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Use ZZ/(2)[t,x,y];
NC.SetOrdering(<quotes>ELIM</quotes>);  
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NC.SetOrdering("LLEX");  
NC.RingEnv();
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F1 := [[x^2], [y,x]]; -- x^2+yx
Coefficient ring : Q
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F2 := [[x,y], [t,y]]; -- xy+ty
Alphabet : abc
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F3 := [[x,t], [t,x]]; -- xt+tx
Ordering : ELIM
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F4 := [[y,t], [t,y]]; -- yt+ty
 +
G := [F1, F2,F3,F4];
 +
Gb:=NC.GB(G);
 +
NC.HF(Gb,5);
  
-------------------------------
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[1, 3, 5, 5, 5, 5]
F1 := [[1,<quotes>a</quotes>],[1,<quotes></quotes>]];
 
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];
 
NC.Add(F1,F2); -- over Q
 
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]
 
-------------------------------
 
NC.SetFp(); -- set default Fp = F2
 
NC.RingEnv();
 
Coefficient ring : Fp = Z/(2)
 
Alphabet : abc
 
Ordering : ELIM
 
 
 
-------------------------------
 
NC.Add(F1,F2); -- over F2
 
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]
 
-------------------------------
 
NC.Add(F1,F1); -- over F2
 
[ ]
 
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NC.Add</see>
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<see>ApCoCoA-1:Use|Use</see>
<see>NC.Deg</see>
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<see>ApCoCoA-1:NC.IsGB|NC.IsGB</see>
<see>NC.FindPolynomials</see>
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<see>ApCoCoA-1:NC.MB|NC.MB</see>
<see>NC.GB</see>
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<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
<see>NC.HF</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
<see>NC.Intersection</see>
 
<see>NC.IsGB</see>
 
<see>NC.KernelOfHomomorphism</see>
 
<see>NC.LC</see>
 
<see>NC.LT</see>
 
<see>NC.NR</see>
 
<see>NC.ReducedGB</see>
 
<see>NC.MRBP</see>
 
<see>NC.SetFp</see>
 
<see>NC.SetOrdering</see>
 
<see>NC.SetRelations</see>
 
<see>NC.SetRules</see>
 
<see>NC.SetX</see>
 
<see>NC.Subtract</see>
 
<see>NC.UnsetFp</see>
 
<see>NC.UnsetOrdering</see>
 
<see>NC.UnsetRelations</see>
 
<see>NC.UnsetRules</see>
 
<see>NC.UnsetX</see>
 
<see>Introduction to CoCoAServer</see>
 
 
</seealso>
 
</seealso>
 
<types>
 
<types>
 
<type>apcocoaserver</type>
 
<type>apcocoaserver</type>
 +
<type>ideal</type>
 
<type>groebner</type>
 
<type>groebner</type>
 +
<type>non_commutative</type>
 
</types>
 
</types>
<key>gbmr.Add</key>
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<key>ncpoly.HF</key>
<key>NC.Add</key>
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<key>NC.HF</key>
<key>Add</key>
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<key>HF</key>
<wiki-category>Package_gbmr</wiki-category>
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<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:34, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.HF

Enumerate the values of the Hilbert-Dehn function of a finitely generated K-algebra.

Syntax

NC.HF(G:LIST[, DB:INT]):LIST

Description

Let P be a finitely generated non-commutative polynomial ring over K, and let I be a finitely generated two-sided ideal in P. Then P/I is a finitely generated K-algebra. Moreover, for every integer i, we let F_{i} be the K-vector subspace generated by the words of length less than or equal to i. Clearly, the set {F_{i}} is a filtration of P. Further, the filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersects I)} of P/I. The Hilbert-Dehn function of K-algebra P/I is a map HF: N --> N defined by HF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I)), i.e. HF(i) is equal to the number of words of length i in a Macaulay's basis (see NC.MB) of P/I.

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Please set non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

  • @param G: a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5 is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial 0 is represented as the empty LIST []. Warning: users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!

  • @return: a LIST of non-negative integers, which are values of the Hilbert-Dehn function of the K-algebra P/<G>.

Optional parameter:

  • @param DB: a positive INT, which is a degree bound of the Hilbert-Dehn function. Note that we set DB=32 by default. Thus, in the case that the K-dimension of P/<G> is finite, it is necessary to set DB to a large enough INT in order to compute all the values of the Hilbert-Dehn function.

Example

Use ZZ/(2)[t,x,y];
NC.SetOrdering("LLEX"); 
F1 := [[x^2], [y,x]]; -- x^2+yx
F2 := [[x,y], [t,y]]; -- xy+ty
F3 := [[x,t], [t,x]]; -- xt+tx
F4 := [[y,t], [t,y]]; -- yt+ty
G := [F1, F2,F3,F4]; 
Gb:=NC.GB(G);
NC.HF(Gb,5);

[1, 3, 5, 5, 5, 5]
-------------------------------

See also

Use

NC.IsGB

NC.MB

NC.SetOrdering

Introduction to CoCoAServer