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

From ApCoCoAWiki
m (fixed links to namespace ApCoCoA)
m (replaced <quotes> tag by real quotes)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
 +
{{Version|1}}
 
<command>
 
<command>
 
<title>NC.HF</title>
 
<title>NC.HF</title>
Line 12: Line 13:
 
<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>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.
+
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>
 
<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>@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>
Line 23: Line 24:
 
<example>
 
<example>
 
Use ZZ/(2)[t,x,y];
 
Use ZZ/(2)[t,x,y];
NC.SetOrdering(<quotes>LLEX</quotes>);  
+
NC.SetOrdering("LLEX");  
 
F1 := [[x^2], [y,x]]; -- x^2+yx
 
F1 := [[x^2], [y,x]]; -- x^2+yx
 
F2 := [[x,y], [t,y]]; -- xy+ty
 
F2 := [[x,y], [t,y]]; -- xy+ty

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