ApCoCoA:Weyl.WMul

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 <command>
   <title>Weyl.WMul</title>
   <short_description>Computes the product F*G of Weyl polynomial F and G in normal form.</short_description>

<syntax> Weyl.WMul(F:POLY,G:POLY):POLY </syntax>

   <description>

This method multiplies two Weyl polynomials F and G and returns F*G as a Weyl polynomial in normal form.

<itemize> <item>@param F A Weyl polynomial.</item> <item>@param G A Weyl polynomial.</item> <item>@return The product F*G in normal form.</item> </itemize>

<example> A1::=QQ[x,d]; -- Define appropriate ring Use A1; F:=x; G:=d; Weyl.WMul(F,G); xd


Weyl.WeylMul(G,F); xd + 1


Weyl.WMul(Weyl.WMul(G,F)-2G,F^3+G); x^4d - 2x^3d + 4x^3 + xd^2 - 6x^2 - 2d^2 + d


-- If you want to multiply Weyl polynomials that are not in normal form say for example F=d^2x^3-2dx^2+7 and G=2d^3x-5xd+3, then -- first convert them into normal form before multiplication.


F:=Weyl.WNormalForm([[d^2,x^3],[-2d,x^2],[7]]); F; x^3d^2 + 4x^2d + 2x + 7


G:=Weyl.WNormalForm([[2d^3,x],[-5x,d],[3]]); G; 2xd^3 - 5xd + 6d^2 + 3


Weyl.WMul(F,G); 2x^4d^5 - 5x^4d^3 + 18x^3d^4 - 27x^3d^2 + 36x^2d^3 + 14xd^3 - 18x^2d + 12xd^2 - 35xd + 42d^2 + 6x + 21


Weyl.WMul(G,F); 2x^4d^5 - 5x^4d^3 + 32x^3d^4 - 32x^3d^2 + 148x^2d^3 + 14xd^3 - 38x^2d + 216xd^2 - 35xd + 42d^2 - 4x + 72d + 21


Weyl.WMul(Weyl.WNormalForm([[d^2,x^3],[-2d,x^2],[7]]),Weyl.WNormalForm([[2d^3,x],[-5x,d],[3]])); 2x^4d^5 - 5x^4d^3 + 18x^3d^4 - 27x^3d^2 + 36x^2d^3 + 14xd^3 - 18x^2d + 12xd^2 - 35xd + 42d^2 + 6x + 21


</example> <example> A2::=ZZ/(101)[x[1..2],y[1..2]]; Use A2; Weyl.WMul(x[1]^11,y[1]^11); x[1]^11y[1]^11


Weyl.WMul(y[1]^11,x[1]^11); x[1]^11y[1]^11 + 20x[1]^10y[1]^10 - 10x[1]^9y[1]^9 + 33x[1]^8y[1]^8 + 23x[1]^7y[1]^7 - 17x[1]^6y[1]^6 - x[1]^5y[1]^5 - 18x[1]^4y[1]^4 - 36x[1]^3y[1]^3 - 36x[1]^2y[1]^2 + 26x[1]y[1] - 16


Weyl.WMul(y[1]^21,x[1]^21); x[1]^21y[1]^21 + 37x[1]^20y[1]^20 + 27x[1]^19y[1]^19 + 17x[1]^18y[1]^18 - 37x[1]^17y[1]^17 + 43x[1]^16y[1]^16 - 17x[1]^15y[1]^15 - 27x[1]^14y[1]^14 - 5x[1]^13y[1]^13 - 49x[1]^12y[1]^12 - 39x[1]^11y[1]^11 - 25x[1]^10y[1]^10 - 40x[1]^9y[1]^9 + 46x[1]^8y[1]^8 - 35x[1]^7y[1]^7 - 47x[1]^6y[1]^6 - 30x[1]^5y[1]^5 + 45x[1]^4y[1]^4 + 40x[1]^3y[1]^3 + 3x[1]^2y[1]^2 + 41x[1]y[1] + 26


--this computation is too slow with Weyl.WeylMul(F,G) </example>

  </description>
   <seealso>
     <see>Weyl.WNormalForm</see>
   </seealso>
   <types>
     <type>polynomial</type>
   </types>
   <key>weyl.wmul</key>
   <key>wmul</key>
   <wiki-category>Package_weyl</wiki-category>
 </command>