Getting Started with Kronecker 0.166-9

Grégoire Lecerf
Laboratoire de mathématiques, LAMA - UMR 8100
Université de Versailles St-Quentin-en-Yvelines
45, avenue des Etats-Unis
78035 Versailles, France

Kronecker is a package for MAGMA to solve polynomial systems of equations. It is the result of a long term research by many people organised around the TERA project.

The present package has been designed by M. Giusti, G. Lecerf and B. Salvy. It is written in Magma by G. Lecerf and also contains contributions of E. Schost and L. Lehmann.

In order to use Kronecker you need a version of Magma at least 2.9-2.

What is "spec"?

In this document "spec" denotes the spec file for loading the Kronecker package. You need to know the full path directory where it is located. For instance if you are at MEDICIS this is "/usr/local/kronecker/spec". Note that this installation must be architecture dependent.

1   How to Solve your System?

Assume you have a system of s polynomial equations and s' inequations in n variables over a field K either of characteristic 0 or big enough:
f1(x1,...,xn) = 0,..., fs(x1,...,xn) = 0,
h1(x1,...,xn) <> 0,..., hs'(x1,...,xn) <> 0,

The resolution is achieved by issuing the following commands:





Example: Solving the system called cyclic 3:
f1=f2=0,   x<> 0, y<> 0, z<> 0.



Verbose Levels: the following instruction will make Kronecker verbose:
The verbosity level can be specified through an integer between 0 and 5.

Now the variable lf contains the solution of the system: it is a sequence of sequences of Lifting Fibers. Each fiber corresponds to an equidimensional component of the variety solution of the system.

The command PrintLF(lf) pretty prints the resolution, DegreeLF(lf) and DimensionLF(lf) return respectively the degree (the sum of the degrees of the irreducible components) and the dimension (the maximum of the dimension of the irreducible components) of the variety described by lf. The command VerifyLF(lf) returns a boolean telling whether lf is correct or not.

2   Optimization of the Input

In order to get good performances with Kronecker it it important to take care about some simple rules concerning the input system. GeometricSolve([(x1+2x2+3x3)(5x2-7x3)]) is better than GeometricSolve([5x1x2-7x1x3+10x22+x2x3-21x32]).
GeometricSolve([x1+x23,x22+x17+x35+1]) should be replaced by GeometricSolve([x22+(-x23)7+x35+1]).
GeometricSolve([(x1-x2)T-1,x22+x17+7]) must be rewritten GeometricSolve([x22+x17+7],[x1-x2]).

As an experimental tool, a fast way to solve your system with Kronecker is to use the python script kroptimize.

3   Dimension Zero

We now describe the meaning of a zero dimensional lifting fiber. Let lf be a fiber of a zero equidimensional variety. Basically it is a record containing the following fields: The set of points described by lf is composed of the points
for all the possible values of i and all the roots u of Q[i](u)=0. Note that P[i] is prime with Q[i] so that the denominator does not vanish over the roots of Q[i].
Example (continued):
> [ DegreeLF(z) : z in lf];
[ 6, 0, 0 ]
> lf0:=lf[1][1];
> lf0`PrimitiveElement;
y + 2*z
> lf0`MinimalPolynomial;
   T^6 + 27
> lf0`Denominator;
> lf0`Parametrization;
        9*T^3 - 81
The system has 6 isolated solutions.

4   Positive Dimension

Let lf be a lifting fiber of a positive dimensional component V of dimension r, some more fields are significative: Let y1,...,yn be new variables, defined by y=M-1(x-b), and fiy be the expression of fi with respect to the yj, and let p be the projection map from V to the affine space spanned by y1,...,yr the following properties hold: The resolution contains a few other fields, consult the reference manual of Kronecker to know more about it.

5   MathML Interface

> KroneckerSetMathMLVerbose();
> lf:=GeometricSolve(equations);
Serving Magma HTTP on port 8000 ...
Then you can use your favorite MathML browser with url: http://localhost:8000/index.html. Use the reload button in order to follow your computations. Once finished, do not forget to kill the web server using the shell command pkill mml4mgm for instance.

Go to the Kronecker home page

This document was translated from LATEX by HEVEA and HACHA.