Difference between revisions of "ApCoCoA-1:BBSGen.NonStandPoly"
From ApCoCoAWiki
(Removing all content from page) |
m (insert version info) |
||
| (11 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
| + | {{Version|1}} | ||
| + | <command> | ||
| + | <title>BBSGen.NonStandPoly</title> | ||
| + | <short_description>This function computes the non-standard polynomial generators of the vanishing ideal of border basis | ||
| + | scheme with respect to the arrow grading. | ||
| + | |||
| + | </short_description> | ||
| + | |||
| + | <syntax> | ||
| + | BBSGen.NonStandPoly(OO,BO,W,N); | ||
| + | BBSGen.NonStandPoly(OO:LIST,BO:LIST,W:MATRIX,N:INTEGER):LIST | ||
| + | </syntax> | ||
| + | <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> | ||
| + | <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>@return List of polynomials and their degree with respect to the arrow grading.</item> | ||
| + | </itemize> | ||
| + | |||
| + | |||
| + | <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)]] | ||
| + | |||
| + | |||
| + | |||
| + | </example> | ||
| + | |||
| + | </description> | ||
| + | <types> | ||
| + | <type>borderbasis</type> | ||
| + | <type>list</type> | ||
| + | <type>apcocoaserver</type> | ||
| + | </types> | ||
| + | |||
| + | <see>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</see> | ||
| + | <see>ApCoCoA-1:BBSGen.NonStand|BBSGen.NonStand</see> | ||
| + | <key>NonStandPoly</key> | ||
| + | <key>BBSGen.NonStandPoly</key> | ||
| + | <key>bbsmingensys.NonStandPoly</key> | ||
| + | <wiki-category>ApCoCoA-1:Package_bbsmingensys</wiki-category> | ||
| + | </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)]]
