ApCoCoA-1:Weyl.BFs
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Weyl.BFs
Computes B-function of a polynomial F in Weyl algebra A_n.
Syntax
Weyl.BFs(F:POLY):POLY
Description
Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.
Computes Bernstein-Sato polynomial (global B-function) of a polynomial F in Weyl algebra A_n. F should not involve any of the indeterminates in {y1,...,yn}. The resulting monic polynomial belongs to the the ring QQ[s].
@param F A polynomial F in the indeterminates x1,...,xn of a Weyl Algebra A_n.
@return A monic polynomial in s of the ring Q[s].
Example
A2::=QQ[x[1..2],d[1..2]]; --Define appropriate ring Use A2; F:=x[1]^3-x[2]^2; ------------------------------- Bf:=Weyl.BFs(F); Computing Annihilating ideal of F . . . . . -- CoCoAServer: computing Cpu Time = 0.079 ------------------------------- Now computing Bernstein-Sato polynomial of F using algorithm of Oaku and Takayama . . . . . -- CoCoAServer: computing Cpu Time = 0.015 ------------------------------- Polynomial of the ring R_s = QQ[s] ------------------------------- Bf; R_s :: 36s^3 + 108s^2 + 107s + 35 ------------------------------- Using R_s Do Factor(Bf); EndUsing; [[s + 1, 1], [6s + 5, 1], [6s + 7, 1]] -------------------------------
Example
A3::=QQ[x[1..3],d[1..3]]; --Define appropriate ring Use A3; F:=x[2]^2-x[1]x[3]-1; Bf:=Weyl.BFs(F); Computing Annihilating ideal of F . . . . . -- CoCoAServer: computing Cpu Time = 0.141 ------------------------------- Now computing Bernstein-Sato polynomial of F using algorithm of Oaku and Takayama . . . . . -- CoCoAServer: computing Cpu Time = 0.031 ------------------------------- Polynomial of the ring R_s = QQ[s] ------------------------------- Bf; R_s :: s + 1 ------------------------------- Using R_s Do Factor(Bf); EndUsing; [[s + 1, 1]] ------------------------------- Weyl.BFs(F^3); Computing Annihilating ideal of F . . . . . -- CoCoAServer: computing Cpu Time = 6.438 ------------------------------- Now computing Bernstein-Sato polynomial of F using algorithm of Oaku and Takayama . . . . . -- CoCoAServer: computing Cpu Time = 0.219 ------------------------------- Polynomial of the ring R_s = QQ[s] R_s :: 9s^3 + 18s^2 + 11s + 2 ------------------------------- Weyl.BFs(x[1]^6-x[2]^5); Computing Annihilating ideal of F . . . . . -- CoCoAServer: computing Cpu Time = 0.563 ------------------------------- Now computing Bernstein-Sato polynomial of F using algorithm of Oaku and Takayama . . . . . -- CoCoAServer: computing Cpu Time = 2.984 ------------------------------- Polynomial of the ring R_s = QQ[s] R_s :: 16815125390625000000000000s^21 + 353117633203125000000000000s^20 + 3513427033007812500000000000s^19 + 22026880088085937500000000000s^18 + 97610575907409960937500000000s^17 + 325093985638078710937500000000s^16 + 844634788534270195312500000000s^15 + 1753734956224006054687500000000s^14 + 2957073443247513199746093750000s^13 + 4091893318114706362324218750000s^12 + 4676334607651372763220703125000s^11 + 4426654391075569515662109375000s^10 + 3470234984704668473597056640625s^9 + 2245607069062054196103978515625s^8 + 1191490565958538530952914843750s^7 + 512754346260873991052388281250s^6 + 176058088164619553236167315625s^5 + 47061818549243955830485015625s^4 + 9431509246910811545973345000s^3 + 1332126841248420215336222500s^2 + 118148969513199005239642784s + 4945056144855740632502784 ------------------------------- Using R_s Do Factor(It);EndUsing; [[s + 1, 1], [15s + 13, 1], [15s + 11, 1], [15s + 19, 1], [15s + 17, 1], [15s + 14, 1], [10s + 11, 1], [30s + 23, 1], [30s + 29, 1], [10s + 9, 1], [15s + 16, 1], [30s + 17, 1], [10s + 13, 1], [15s + 22, 1], [30s + 49, 1], [30s + 31, 1], [15s + 8, 1], [30s + 37, 1], [30s + 43, 1], [30s + 11, 1], [10s + 7, 1]] -------------------------------
See also
