Difference between revisions of "ApCoCoA-1:BBSGen.TraceSyzFull"
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(New page: <command> <title>BBSGen.TraceSyzFull</title> <short_description>: Let Tau^kl_ij :=t[k,l,i,j] be the (i,j) ^th entry of matrix the operation [A_k,A_l]. The result of the Trace Syzyg...) |
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| + | {{Version|1}} | ||
<command> | <command> | ||
<title>BBSGen.TraceSyzFull</title> | <title>BBSGen.TraceSyzFull</title> | ||
| − | <short_description> | + | <short_description>This function computes the trace polynomials. |
| − | |||
| Line 12: | Line 12: | ||
</syntax> | </syntax> | ||
<description> | <description> | ||
| + | Let l,k_1,....,k_s in {1,...,n} where s is a positive integer and | ||
| + | Pi=x_{k_1}...x_{k_s}x_l | ||
| + | a term (or power product) from the polynomial ring K[x_1,...,x_N]. Let the generic multiplication matrices A_{k_1},...,A_{k_s},A_l in Mat(K[c]) be associated to the indeterminates in Pi. We shall name the polynomial | ||
| + | Trace([A_{k_1}...A_{k_s},A_l]) in K[c] | ||
| + | as the trace polynomial with respect to Pi and variable x_l. We shall denote it by | ||
| + | T_{Pi,x_l}. | ||
| + | This function computes every trace polynomial with respect to every Pi with log that is equal to a non-standard degree of an element from tau and with respect to every variable from {x_1,...,x_N}. | ||
| + | <par/> | ||
| + | NOTE: This function due to the growth of polynomials during the matrix multiplication, may not give result for every ring and order ideal. In that case we recommend <ref>ApCoCoA-1:BBSGen.TraceSyzStep|BBSGen.TraceSyzStep</ref> and | ||
| + | <ref>ApCoCoA-1:BBSGen.TraceSyzLin|BBSGen.TraceSyzLin</ref> . | ||
| + | |||
| + | |||
| + | |||
<itemize> | <itemize> | ||
| − | <item>@param The order ideal OO, border BO, the number of | + | <item>@param The order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N]. |
</item> | </item> | ||
<item>@return The list of Trace Syzygy polynomials. </item> | <item>@return The list of Trace Syzygy polynomials. </item> | ||
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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); | ||
Mu:=Len(OO); | Mu:=Len(OO); | ||
Nu:=Len(BO); | Nu:=Len(BO); | ||
| − | + | ||
Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]; | Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]; | ||
| Line 56: | Line 70: | ||
</description> | </description> | ||
<types> | <types> | ||
| − | <type> | + | <type>bbsmingensys</type> |
| − | <type> | + | <type>list</type> |
<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||
</types> | </types> | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | <key> | + | <see>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</see> |
| − | <key>BBSGen. | + | <see>ApCoCoA-1:BBSGen.TraceSyzStep|BBSGen.TraceSyzStep</see> |
| − | <key>bbsmingensys. | + | <see>ApCoCoA-1:BBSGen.TraceSyzLin|BBSGen.TraceSyzLin</see> |
| − | <wiki-category>Package_bbsmingensys</wiki-category> | + | |
| + | <key>TraceSyzFull</key> | ||
| + | <key>BBSGen.TraceSyzFull</key> | ||
| + | <key>bbsmingensys.TraceSyzFull</key> | ||
| + | <wiki-category>ApCoCoA-1:Package_bbsmingensys</wiki-category> | ||
</command> | </command> | ||
Latest revision as of 09:51, 7 October 2020
| This article is about a function from ApCoCoA-1. |
BBSGen.TraceSyzFull
This function computes the trace polynomials.
Syntax
TraceSyzFull(OO,BO,N); TraceSyzFull(OO:LIST,BO:LIST,N:INTEGER):LIST
Description
Let l,k_1,....,k_s in {1,...,n} where s is a positive integer and
Pi=x_{k_1}...x_{k_s}x_l
a term (or power product) from the polynomial ring K[x_1,...,x_N]. Let the generic multiplication matrices A_{k_1},...,A_{k_s},A_l in Mat(K[c]) be associated to the indeterminates in Pi. We shall name the polynomial
Trace([A_{k_1}...A_{k_s},A_l]) in K[c]
as the trace polynomial with respect to Pi and variable x_l. We shall denote it by
T_{Pi,x_l}.
This function computes every trace polynomial with respect to every Pi with log that is equal to a non-standard degree of an element from tau and with respect to every variable from {x_1,...,x_N}.
NOTE: This function due to the growth of polynomials during the matrix multiplication, may not give result for every ring and order ideal. In that case we recommend BBSGen.TraceSyzStep and
@param The order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].
@return The list of Trace Syzygy polynomials.
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); Mu:=Len(OO); Nu:=Len(BO); Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]; BBSGen.TraceSyzFull(OO,BO,N); [c[1,2]t[1,2,3,1] + c[2,2]t[1,2,3,2] + c[3,2]t[1,2,3,3] + c[4,2]t[1,2,3,4] + c[1,4]t[1,2,4,1] + c[2,4]t[1,2,4,2] + c[3,4]t[1,2,4,3] + c[4,4]t[1,2,4,4] + t[1,2,1,3] + t[1,2,2,4], 2c[1,1]t[1,2,2,1] + 2c[2,1]t[1,2,2,2] + 2c[3,1]t[1,2,2,3] + 2c[4,1]t[1,2,2,4]+ 2c[1,3]t[1,2,4,1] + 2c[2,3]t[1,2,4,2] + 2c[3,3]t[1,2,4,3] + 2c[4,3]t[1,2,4,4] + 2t[1,2,1,2] + 2t[1,2,3,4], c[1,2]t[1,2,3,1] + c[2,2]t[1,2,3,2] + c[3,2]t[1,2,3,3] + c[4,2]t[1,2,3,4] + c[1,4]t[1,2,4,1] + c[2,4]t[1,2,4,2] + c[3,4]t[1,2,4,3] + c[4,4]t[1,2,4,4] + t[1,2,1,3] + t[1,2,2,4], 2c[1,2]c[3,1]t[1,2,2,1] + 2c[1,4]c[4,1]t[1,2,2,1] + 2c[2,2]c[3,1]t[1,2,2,2] + 2c[2,4]c[4,1]t[1,2,2,2] + 2c[3,1]c[3,2]t[1,2,2,3] + 2c[3,4]c[4,1]t[1,2,2,3] + 2c[3,1]c[4,2]t[1,2,2,4] + 2c[4,1]c[4,4]t[1,2,2,4] + 2c[1,2]c[3,3]t[1,2,4,1] + 2c[1,4]c[4,3]t[1,2,4,1] + 2c[2,2]c[3,3]t[1,2,4,2] + 2c[2,4]c[4,3]t[1,2,4,2] + 2c[3,2]c[3,3]t[1,2,4,3] + 2c[3,4]c[4,3]t[1,2,4,3] + 2c[3,3]c[4,2]t[1,2,4,4] + 2c[4,3]c[4,4]t[1,2,4,4] + 2c[1,1]t[1,2,2,3] + 2c[2,1]t[1,2,2,4] + 2c[1,4]t[1,2,3,1] + 2c[2,4]t[1,2,3,2] + 2c[3,4]t[1,2,3,3] + 2c[4,4]t[1,2,3,4] + 2c[1,3]t[1,2,4,3] + 2c[2,3]t[1,2,4,4] + 2t[1,2,1,4]]
