# A Maple package for locally and globally studying systems of pseudo linear equations.

Authors: M. A Barkatou, T. Cluzeau, and A. El Hajj

I - Introduction

Pseudo-linear systems constitute a large class of linear functional systems including the usual differential, difference and q-difference systems.
The Maple package PseudoLinearSystems is dedicated to the study of this class of linear systems. It contains a generic procedure for computing the so-called simple form of a pseudo-linear system as well as local data useful for the local analysis.
It is also devoted for the computation of rational solutions (using simple forms) of a single linear differential, difference or q-difference system, and rational solutions of a system of mixed linear partial differential, difference or q-difference equations.
The PseudoLinearSystems package contains implementations of the algorithms developped in our accepted paper:
"Simple Forms and Rational Solutions of Pseudo-Linear Systems".

To install it, you must proceed as follows:
1. Copy the previous .m file in a directory called "PseudoLinearSystems"
2. Add this directory to your libname, for example by performing the two following steps:

• Open Maple and type

libname;

• Then, type

libname := " the global path of the directory
", the result of the previous step;

• After that

read "the global path of the directory
/PseudoLinearSystems.m";
3. Type

with(PseudoLinearSystems);

You must get the list of the functions contained in the package. If you do not manage to install the package, then contact us.

III - List of main procedures

Here are the main procedures of our package with some examples files illustrating how to use them.

1. SimpleForm

This is a generic procedure to compute a simple form of the pseudo linear system L(Y)=A\delta(Y)+B\phi(Y)=0, where A and B are matrices with rational function entries admitting power series expansion in some local parameter t. Here \phi is an automorphism preserving the valuation, and \delta is a \phi-derivation, and they are always given as inputs.
The output is a list containing respectively the matrices newA and newB of the equivalent simple system newL(Y)=newA.\delta(Y)+newB.\phi(Y), the two invertible matrices S and T such that newL =SLT, and the determinant of the new leading matrix pencil thqt is thus not zero.

• Example of a differential system written as A\delta(Y)+B\phi(Y)=0: Maple file   PDF file

• Example of a difference system written as A\delta(Y)+B\phi(Y)=0: Maple file   PDF file

• Example of a q-difference system written as A\delta(Y)+B\phi(Y)=0: Maple file   PDF file

• Example of a pseudo-linear system A\delta(Y)+B\phi(Y)=0 at infinity with \phi defined as \phi(x)=ax+b : Maple file   PDF file

2. K_SimpleForm

This procedure computes k-simple forms of the pseudo linear system \delta(Y)=M\phi(Y), where M is a matrix with Laurent series entries in some local parameter t. Here \phi is an automorphism preserving the valuation, and \delta is a \phi-derivation.
The output is the matrix of the equivalent simple system, the transformation matrix, and the correspnding characteristic polynomial.

3. SuperReduced

This procedure computes a super-irreducible form, based on simple forms, of the pseudo linear system \delta(Y)=M\phi(Y), where M is a matrix with Laurent series entries in some local parameter t. Here \phi is an automorphism preserving the valuation, and \delta is a \phi-derivation.
The output is the matrix of the equivalent simple system, the list of the characteristic polynomials of each of the k-simple systems for k = m...0, where m is the minimal Poincare rank, and the transformation matrix.

• Example of a differential system written as \delta(Y)=M\phi(Y): Maple file   PDF file

• Example of a difference system written as \delta(Y)=M\phi(Y): Maple file   PDF file

• Example of a q-difference system written as \delta(Y)=M\phi(Y): Maple file   PDF file

• Example of a pseudo-linear system \delta(Y)=M\phi(Y) at infinity with \phi defined as \phi(x)=ax+b : Maple file   PDF file

4. SuperReduced_FirstOrder

This procedure computes a super-irreducible form, based on simple forms, of a first order differential system (Y'=MY), difference system (Y(x+1)=MY), and q-difference system (Y(qx)=MY), where M is a matrix with Laurent series entries in some local parameter t.
The output is the matrix of the equivalent simple system, the list of the characteristic polynomials of each of the k-simple systems for k = m...0, where m is the minimal Poincare rank, and the transformation matrix.

5. MinimalPoincareRank

This procedure computes the minimal Poincare rank, based on simple forms, of the pseudo linear system \delta(Y)=M\phi(Y), where M is a matrix with Laurent series entries in some local parameter t. Here \phi is an automorphism preserving the valuation, and \delta is a \phi-derivation.
The output is the minimal Poincare rank, the equivalent matrix, the correspnding characteristic polynomial, and the transformation matrix.

6. MinimalPoincareRank_FirstOrder

This procedure computes the minimal Poincare rank, based on simple forms, of a first order differential system (Y'=MY), difference system (Y(x+1)=MY), and q-difference system (Y(qx)=MY), where M is a matrix with Laurent series entries in some local parameter t.
The output is the minimal Poincare rank, the equivalent matrix, the correspnding characteristic polynomial, and the transformation matrix.

7. IntegerSlopesNewtonPolygon

This procedure computes the integer slopes of the Newton Polygon, based on simple forms, of the pseudo linear system \delta(Y)=M\phi(Y), where M is a matrix with Laurent series entries in some local parameter t.
Here \phi is an automorphism preserving the valuation, and \delta is a \phi-derivation. The output are the integer slopes of the Newton Polygon, and the correspnding characteristic polynomials.

• Example of a pseudo-linear system \delta(Y)=M\phi(Y) at infinity with \phi defined as \phi(x)=ax+b : Maple file   PDF file

8. IntegerSlopesNewtonPolygon_FirstOrder

This procedure computes the integer slopes of the Newton Polygon, based on simple forms, of a first order differential system (Y'=MY), difference system (Y(x+1)=MY), and q-difference system (Y(qx)=MY), where M is a matrix with Laurent series entries in some local parameter t.
The output are the integer slopes of the Newton Polygon, and the correspnding characteristic polynomials.

9. RationalSolutions_1System

This procedure computes a basis of all rational solutions,based on simple forms, of a first order differential system (Y'=MY), difference system (Y(x+1)=MY), and q-difference system (Y(qx)=MY), where M is a matrix with rational function entries.
It returns {} if there is no non-trivial rational solutions.

10. PolynomialSolutions_1System

This procedure computes a basis of all polynomial solutions, based on simple forms, of a first order differential system (Y'=MY), difference system (Y(x+1)=MY), and q-difference system (Y(qx)=MY), where M is a matrix with rational function entries.
It returns {} if there is no non-trivial polynomial solutions.

11. RationalSolutions

This procedure computes a basis of all rational solutions of a fully integrable system composed of mixed first order differential, difference and q-difference equations.
It returns {} if there is no non-trivial rational solutions.

12. PolynomialSolutions

This procedure computes a basis of all polynomial solutions of a fully integrable system composed of mixed first order differential, difference and q-difference equations.
It returns {} if there is no non-trivial polynomial solutions.

• Example of mixed differential, difference and q-difference equations: Maple file   PDF file

13. Eigenring

This procedure computes the Eigenring of a fully integrable system composed of mixed first order differential, difference and q-difference equations.

14. Decompose

This procedure decomposes a fully integrable system of mixed first order differential, difference and q-difference equations, into an equivalent system having a block diagonal form (Using the Eigenring).
The output is a list containing the equivalent block diagonal matrices, the corresponding variables, and the corresponfing types of the systems (difference, q-difference, or differential).

• Example of mixed differential, difference and q-difference equations: Maple file   PDF file