Difference between revisions of "ApCoCoA-1:BBSGen.NonStandPoly"
From ApCoCoAWiki
| Line 1: | Line 1: | ||
<command> | <command> | ||
<title>BBSGen.Wmat</title> | <title>BBSGen.Wmat</title> | ||
| − | <short_description>This function computes the non-standard | + | <short_description>This function computes the non-standard polynomial generators of the vanishing ideal of border basis |
| − | scheme. | + | scheme with respect to the arrow grading. |
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
| − | NonStandPoly(OO,BO,W,N); | + | BBSGen.NonStandPoly(OO,BO,W,N); |
| − | NonStandPoly(OO:LIST,BO:LIST,W:MATRIX,N:INTEGER):LIST | + | BBSGen.NonStandPoly(OO:LIST,BO:LIST,W:MATRIX,N:INTEGER):LIST |
</syntax> | </syntax> | ||
| − | <description> | + | <description>Let W be the weight matrix with respect to the arrow grading.(see <ref>BBSGen.Wmat</ref>) |
| − | + | Let \tau^kl_ij be a polynomials from the generating set of the vanishing ideal of border basis scheme. It is called standard, if deg_W(\tau^kl_ij) has exactly one strictly positive component. If \tau^kl_ij is not standard then it is called non-standard. This function computes such non-standard polynomials. | |
<itemize> | <itemize> | ||
<item>@param The order ideal OO, BO border of OO , the number of indeterminates of the polynomial ring N and the Weight Matrix. | <item>@param The order ideal OO, BO border of OO , the number of indeterminates of the polynomial ring N and the Weight Matrix. | ||
</item> | </item> | ||
| − | <item>@return List of polynomials and their degree | + | <item>@return List of polynomials and their degree with respect to the arrow grading. .</item> |
</itemize> | </itemize> | ||
| Line 25: | Line 25: | ||
OO:=BB.Box([1,1]); | OO:=BB.Box([1,1]); | ||
BO:=BB.Border(OO); | BO:=BB.Border(OO); | ||
| + | N:=Len(Indets()); | ||
W:=BBSGen.Wmat(OO,BO,N); | W:=BBSGen.Wmat(OO,BO,N); | ||
XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]; | XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]; | ||
Use XX; | Use XX; | ||
| + | |||
BBSGen.NonStandPoly(OO,BO,W,N); | BBSGen.NonStandPoly(OO,BO,W,N); | ||
| Line 58: | Line 60: | ||
<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||
</types> | </types> | ||
| − | + | ||
| − | |||
<see>BBSGen.Wmat</see> | <see>BBSGen.Wmat</see> | ||
<see>BBSGen.NonStand</see> | <see>BBSGen.NonStand</see> | ||
Revision as of 09:53, 8 June 2012
BBSGen.Wmat
This function computes the non-standard polynomial generators of the vanishing ideal of border basis
scheme with respect to the arrow grading.
Syntax
BBSGen.NonStandPoly(OO,BO,W,N); BBSGen.NonStandPoly(OO:LIST,BO:LIST,W:MATRIX,N:INTEGER):LIST
Description
Let W be the weight matrix with respect to the arrow grading.(see BBSGen.Wmat)
Let \tau^kl_ij be a polynomials from the generating set of the vanishing ideal of border basis scheme. It is called standard, if deg_W(\tau^kl_ij) has exactly one strictly positive component. If \tau^kl_ij is not standard then it is called non-standard. This function computes such non-standard polynomials.
@param The order ideal OO, BO border of OO , the number of indeterminates of the polynomial ring N and the Weight Matrix.
@return List of polynomials and their degree with respect to the arrow grading. .
Example
Use R::=QQ[x[1..2]];
OO:=BB.Box([1,1]);
BO:=BB.Border(OO);
N:=Len(Indets());
W:=BBSGen.Wmat(OO,BO,N);
XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];
Use XX;
BBSGen.NonStandPoly(OO,BO,W,N);
[ c[1,2]c[3,1] + c[1,4]c[4,1] - c[1,3],
R :: Vector(1, 2)],
[ c[1,1]c[2,2] + c[1,3]c[4,2] - c[1,4],
R :: Vector(2, 1)],
[ c[1,1]c[2,4] - c[1,2]c[3,3] - c[1,4]c[4,3] + c[1,3]c[4,4],
R :: Vector(2, 2)],
[c[2,2]c[3,1] + c[2,4]c[4,1] - c[2,3],
R :: Vector(1, 1)],
[c[2,1]c[2,4] - c[2,2]c[3,3] - c[2,4]c[4,3] + c[2,3]c[4,4] + c[1,4],
R :: Vector(2, 1)],
[c[2,2]c[3,1] + c[3,3]c[4,2] - c[3,4],
R :: Vector(1, 1)],
[c[2,4]c[3,1] - c[3,2]c[3,3] - c[3,4]c[4,3] + c[3,3]c[4,4] - c[1,3],
R :: Vector(1, 2)],
[c[2,4]c[4,1] - c[3,3]c[4,2] - c[2,3] + c[3,4],
R :: Vector(1, 1)]]
