Difference between revisions of "ApCoCoA-1:BBSGen.NonStandPoly"
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| + | {{Version|1}} | ||
<command> | <command> | ||
| − | <title>BBSGen. | + | <title>BBSGen.NonStandPoly</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>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</ref>). |
| − | + | Let tau^kl_ij be a polynomials from the generating set Tau 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 | + | <item>@param The order ideal OO, BO border of OO , the number of indeterminates of the polynomial ring K[x_1,...,x_N] and the weight matrix(<ref>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</ref>). |
</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 23: | Line 24: | ||
Use R::=QQ[x[1..2]]; | Use R::=QQ[x[1..2]]; | ||
| − | OO:= | + | OO:=$apcocoa/borderbasis.Box([1,1]); |
| − | BO:= | + | BO:=$apcocoa/borderbasis.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 55: | Line 58: | ||
<types> | <types> | ||
<type>borderbasis</type> | <type>borderbasis</type> | ||
| − | <type> | + | <type>list</type> |
<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||
</types> | </types> | ||
| − | + | ||
| − | + | <see>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</see> | |
| − | + | <see>ApCoCoA-1:BBSGen.NonStand|BBSGen.NonStand</see> | |
| − | <see>BBSGen.NonStand</see> | + | <key>NonStandPoly</key> |
| − | <key> | + | <key>BBSGen.NonStandPoly</key> |
| − | <key>BBSGen. | + | <key>bbsmingensys.NonStandPoly</key> |
| − | <key>bbsmingensys. | + | <wiki-category>ApCoCoA-1:Package_bbsmingensys</wiki-category> |
| − | <wiki-category>Package_bbsmingensys</wiki-category> | ||
</command> | </command> | ||
Latest revision as of 09:50, 7 October 2020
| This article is about a function from ApCoCoA-1. |
BBSGen.NonStandPoly
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 Tau 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 K[x_1,...,x_N] and the weight matrix(BBSGen.Wmat).
@return List of polynomials and their degree with respect to the arrow grading.
Example
Use R::=QQ[x[1..2]];
OO:=$apcocoa/borderbasis.Box([1,1]);
BO:=$apcocoa/borderbasis.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)]]
