# ApCoCoA-1:Bertini.BSolve

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## Bertini.BSolve

Solves a zero dimensional homogeneous or non-homogeneous polynomial system of equations with default configurations.

### Syntax

```Bertini.BSolve(P:LIST, SysTyp:STRING)
```

### 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.

This function uses total degree homotopy to find all isolated solutions of a zero dimensional system of polynomial equations. It uses default configurations provided by Bertini. The system of polynomials may be homogeneous or nonhomogeneous. For homogeneous polynomial system the output will be the list of all real solutions and for nonhomogeneous system the output will be the list of all finite solutions.

• @param P: List of polynomials of the given system.

• @param SysTyp: Type of polynomials in the list P. Homogeneous ("hom") or nonhomogeneous ("Nhom").

• @return A list of lists containing the finite (or real) solutions of the system P.

#### Example

```-- An example of zero dimensional Non-Homogeneous Solving.
-- We want to solve zero dimensional non-homogeneous system x^2+y^2-5=0, xy-2=0.

Use S ::= QQ[x,y];
P := [x^2+y^2-5, xy-2];
SysTyp := "Nhom";

-- Then we compute the solution with
Bertini.BSolve(P,SysTyp);

-- And we achieve a list of lists containing all finite solutions.
----------------------------------------
[
[
Vector(400000000000003/200000000000000, -3416759775755413/500000000000000000000000000000),
Vector(9999999999999927/10000000000000000, 8966048861359829/1000000000000000000000000000000)
],
[
Vector(2499999999999963/2500000000000000, 5007041073746771/100000000000000000000000000000),
Vector(249999999999999/125000000000000, -1089183184148021/25000000000000000000000000000)
],
[
Vector(-9999999999999969/10000000000000000, 191792591213411/125000000000000000000000000000),
Vector(-1999999999999999/1000000000000000, 2443331461729629/2500000000000000000000000000000)
],
[
Vector(-250000000000001/125000000000000, 4347064  850996171/1000000000000000000000000000000),
Vector(-9999999999999943/10000000000000000, -2154842536286333/500000000000000000000000000000)
]
]

--The elements of lists are vectors. Each vector represents a complex number. For example Vector(5000/1000,-4150/1000)
--represents the complex number 5000/1000-4150/1000i.
--For Bertini output files please refer to ApCoCoA directory/Bertini.
```

#### Example

```-- An example of zero dimensional Homogeneous Solving
-- We want to solve zero dimensional homogeneous system x^2-z^2=0, xy-z^2=0.

Use S ::= QQ[x,y,z];
M := [x^2-z^2, xy-z^2];
SysTyp := "hom";

-- Then we compute the solution with
Bertini.BSolve(M,SysTyp);

-- And we achieve a list of lists containing all real solutions.
----------------------------------------
[
[2190685167348543/5000000000000000, 2190685167348543/5000000000000000, 2190685167348543/5000000000000000],
[1237092982347763/5000000000000000, 1237092982347763/5000000000000000, -1237092982347763/5000000000000000],
[3235177805819999/100000000000000000000000000000, 9932123317905381/10000000000000000,621807549382663/5000000000000000000000000000],
[3006769352985381/100000000000000000000000000000,1241515414738241/1250000000000000, 555981798431817/5000000000000000000000000000]
]

--These are the real solutions of the system
--For Bertini output files please refer to ApCoCoA directory/Bertini.
------------------------------------
```