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

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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>NC.MB</ref>) of <tt>P/I</tt>.
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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/>
 
<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 non-commutative polynomial ring (via the command <ref>Use</ref>) and word ordering (via the function <ref>NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant commands and 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 (<quotes>LLEX</quotes>). For more information, please check the relevant commands and functions.
 
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<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>@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>
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</description>
 
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<seealso>
 
<seealso>
<see>Use</see>
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<see>ApCoCoA-1:Use|Use</see>
<see>NC.IsGB</see>
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<see>ApCoCoA-1:NC.IsGB|NC.IsGB</see>
<see>NC.MB</see>
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<see>ApCoCoA-1:NC.MB|NC.MB</see>
<see>NC.SetOrdering</see>
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<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
<see>Introduction to CoCoAServer</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
</seealso>
 
</seealso>
 
<types>
 
<types>

Revision as of 08:22, 7 October 2020

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(<quotes>LLEX</quotes>); 
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