# ApCoCoA-1:Weyl.BFs

 This article is about a function from ApCoCoA-1.

## 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]]
-------------------------------
```