Difference between revisions of "ApCoCoA-1:Weyl.WNormalRemainder"

From ApCoCoAWiki
(New page: <command> <title>Weyl.WNormalRemainder</title> <short_description>Computes the normal remainder of a Weyl polynomial <tt>F</tt> with respect to a polynomial or a set of polynomi...)
 
m (insert version info)
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
   <command>
+
   {{Version|1}}
 +
<command>
 
     <title>Weyl.WNormalRemainder</title>
 
     <title>Weyl.WNormalRemainder</title>
 
     <short_description>Computes the normal remainder of a Weyl polynomial <tt>F</tt> with respect
 
     <short_description>Computes the normal remainder of a Weyl polynomial <tt>F</tt> with respect
Line 8: Line 9:
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
<par/>
 
 
Computes the normal remainder of a Weyl polynomial <tt>F</tt> with respect to a polynomial <tt>G</tt> or a set of polynomials in the list <tt>G</tt>.
 
Computes the normal remainder of a Weyl polynomial <tt>F</tt> with respect to a polynomial <tt>G</tt> or a set of polynomials in the list <tt>G</tt>.
 
If <tt>G</tt> is Groebner basis then this function is used for ideal membership problem.
 
If <tt>G</tt> is Groebner basis then this function is used for ideal membership problem.
Line 18: Line 18:
 
</itemize>
 
</itemize>
  
<em>Note:</em> All polynomials that are not in normal form should be first converted into normal form using <ref>Weyl.WNormalForm</ref>, otherwise you may get unexpected results.
+
<em>Note:</em> All polynomials that are not in normal form should be first converted into normal form using <ref>ApCoCoA-1:Weyl.WNormalForm|Weyl.WNormalForm</ref>, otherwise you may get unexpected results.
  
 
<example>
 
<example>
Line 43: Line 43:
 
   </description>
 
   </description>
 
     <seealso>
 
     <seealso>
       <see>Weyl.WNormalForm</see>
+
       <see>ApCoCoA-1:Weyl.WNormalForm|Weyl.WNormalForm</see>
       <see>Weyl.WNR</see>
+
       <see>ApCoCoA-1:Weyl.WNR|Weyl.WNR</see>
 
     </seealso>
 
     </seealso>
 
     <types>
 
     <types>
Line 53: Line 53:
 
     <key>weyl normal remainder</key>
 
     <key>weyl normal remainder</key>
 
     <key>WNormalRemainder</key>
 
     <key>WNormalRemainder</key>
     <wiki-category>Package_weyl</wiki-category>
+
     <wiki-category>ApCoCoA-1:Package_weyl</wiki-category>
 
   </command>
 
   </command>

Latest revision as of 10:39, 7 October 2020

This article is about a function from ApCoCoA-1.

Weyl.WNormalRemainder

Computes the normal remainder of a Weyl polynomial F with respect

to a polynomial or a set of polynomials.

Syntax

Weyl.WNormalRemainder(F:POLY,G:POLY):POLY
Weyl.WNormalRemainder(F:POLY,G:LIST):POLY

Description

Computes the normal remainder of a Weyl polynomial F with respect to a polynomial G or a set of polynomials in the list G.

If G is Groebner basis then this function is used for ideal membership problem.

  • @param F A Weyl polynomial in normal form.

  • @param G A Weyl polynomial or a list of Weyl polynomials.

  • @return The remainder as a Weyl polynomial using normal remainder algorithm in Weyl algebra A_n.

Note: All polynomials that are not in normal form should be first converted into normal form using Weyl.WNormalForm, otherwise you may get unexpected results.

Example

W3::=ZZ/(7)[x[1..3],d[1..3]];
Use W3;
F1:=-d[1]^3d[2]^5d[3]^5+x[2]^5;
F2:=-3x[2]d[2]^5d[3]^5+x[2]d[1]^3;
F3:=-2d[1]^4d[2]^5-x[1]d[2]^7+x[3]^3d[3]^5;
L:=[F1,F2,F3];
Weyl.WNormalRemainder(F1,L);
0
-------------------------------
Weyl.WNormalRemainder(F1,Gens(Ideal(F2,F3)));
-d[1]^3d[2]^5d[3]^5 + x[2]^5
-------------------------------
Weyl.WNormalRemainder(x[2]^5-d[1]^3,L);
x[2]^5 - d[1]^3
-------------------------------
Weyl.WNormalRemainder(x[2]^5-d[1]^3d[2]^7d[3]^6,F1);
-x[2]^5d[2]^2d[3] - 3x[2]^4d[2]d[3] + x[2]^5 + x[2]^3d[3]
-------------------------------


See also

Weyl.WNormalForm

Weyl.WNR