Title
Padé-Hermite approximants and eigenrings of skew-polynomials. |
Version
française |

**Location**

INRIA, Project
CAFÉ

BP 93, 06902 France

**Information**

**Description**

Skew-polynomials are linear operators that include differential
and difference operators. They are used to represent linear differential,
difference, or more general functional equations.
The *eigenring* of a skew-polynomial *L* in *K[X]*
is the ring of endomorphisms of the quotient module *K[X]/K[X] L*.
That ring is useful in solving the corresponding functional equation
*L(y)=0* since its elements can be used to reduce that
problem to solving equations of lower order. In the general case,
eigenrings are computed via the companion system associated to the
operator, which increases the complexity of the problem to computing
eigenrings of systems. For differential equations, there is an efficient
algorithm that works directly with operators [3].

Padé approximants are rational approximations of formal
power series: a power series *f(z)* in *F[[z]]* is approximated
by a rational function *p(z)/q(z)* where *p(z)* and *q(z)*
are polynomials of bounded degree satisfying *q(z) f(z) = p(z) + O(z^n)*
for an integer *n* large enough. Their generalization to a number
of power series *f1(z),...,fm(z)* is a vector *p1(z),...,pm(z)*
of polynomials such that *p1(z)f1(z)+...+pm(z)fm(z) = O(z^n)* for
*n* large enough. There are fast (quadratic) algorithms for computing
such approximants [1,2].

We have recently developped a new algorithm for the eigenring that
works for skew polynomials without converting them to systems.
This algorithm yields candidates for the endomorphisms that contain
unknown constants. An efficient way to compute those constants would be
to apply those candidates to a basis of formal power series solutions
of the equation *L(y)=0* and compute Padé-Hermite
approximants of the resulting series. The goal of this internship is
therefore to provide an efficient implementation of Padé-Hermite
approximants, and to use them as an alternative in the implementation of
the new eigenring algorithm for both the differential and difference equations.
In addition, a variant based on Hensel-lifting truncated eigenring elements
should be studied, implemented and compared to the Padé-Hermite approach.

[1] B.Beckermann & G.Labahn (1994):
*
A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants*,
SIAM Journal on Matrix Analysis and Applications **15**, pp. 804-823.

[2] H.Derksen (1994):
*
An algorithm to compute generalized Padé-Hermite Forms*,
Report 9403, Dept. of Math., Catholic University Nijmegen.

[3] M.van Hoeij (1996):
Rational solutions of the mixed differential equation and its application to
factorization of differential operators,
Proceedings of ISSAC'96, ACM Press, New York.

**Tools**

Unix workstation, the Aldor programming language, with the Algebra library.

**Duration**

3 months.