# Difference between revisions of "ApCoCoA-1:GLPK.L01PSolve"

## GLPK.L01PSolve

Solve a system of polynomial equations over F_2 for one solution in F_2^n.

### Syntax

```GLPK.L01PSolve(F:LIST, CuttingNumber:INT, QStrategy:INT, CStrategy:INT, MinMax: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 finds one solution in F_2^n of a system of polynomial equations over the field F_2. It operates in two stages. Firstly, it models the problem of finding one solution of given polynomial system into a mixed integer linear programming problem. For this the system is first converted into an equivalent CNF form and then the CNF form is converted into a system of equalities and inequalities. Secondly, the mixed integer linear programming model is solved using glpk.

• @param F: A List containing the polynomials of the given system.

• @param CuttingNumber: Maximal support-length of the linear polynomials for conversion to CNF. The possible value could be from 3 to 6.

• @param QStrategy: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;

• @param CStrategy: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;

• @param MinMax: Optimization direction i.e. minimization ("Min") or maximization ("Max").

#### Example

```Use Z/(2)[x[1..4]];
F:=[
xx + xx + xx + xx + x + x + 1,
xx + xx + xx + xx + x + x + 1,
xx + xx + xx + xx + x + x + 1,
xx + xx + xx + xx + 1
];

CuttingNumber:=6;
QStrategy:=0;
CStrategy:=0;
MinMax:=<quotes>Max</quotes>;

-- Then we compute the solution with

GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);

-- The result will be the following:

Converting to CNF with CuttingLength: 6, QStrategy: Standard, CStrategy: Standard.
Converting CNF to system of equalities and inequalities...
Model is ready to solve with GLPK...
Solution Status: INTEGER OPTIMAL
Value of objective function: 2
[0, 1, 0, 1]
-------------------------------
```

#### Example

```Use S::=Z/(2)[x[1..5]];
F:=[
xx + xx + xx + x + x,
xx + xx + xx + xx + xx + xx + x + x + x + 1,
xx + xx + x + x + x,
xx + xx + xx + x + x + x + x + 1,
xx + xx + xx + xx + xx + x + x + x + x
];

CuttingNumber:=6;
QStrategy:=1;
CStrategy:=0;
MinMax:=<quotes>Max</quotes>;

-- Then we compute the solution with

GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);

-- The result will be the following:

Converting to CNF with CuttingLength: 6, QStrategy: LinearPartner, CStrategy: Standard.
Converting CNF to system of equalities and inequalities...
Model is ready to solve with GLPK...
Solution Status: INTEGER OPTIMAL
Value of objective function: 4
[1, 1, 1, 1, 0]
-------------------------------
```

#### Example

```Use ZZ/(2)[x[1..3]];
F := [ xxx + xx + xx + x + x +1,
xxx + xx + xx + x + x,
xx + xx + x
];

CuttingNumber:=5;
QStrategy:=0;
CStrategy:=1;
MinMax:=<quotes>Max</quotes>;

-- Then we compute the solution with

GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);

-- The result will be the following:

Converting to CNF with CuttingLength: 5, QStrategy: Standard, CStrategy: CubicParnterDegree2.
Converting CNF to system of equalities and inequalities...
Model is ready to solve with GLPK...
Solution Status: INTEGER OPTIMAL
Value of objective function: 1
[0, 0, 1]
-------------------------------
```