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ABSTRACT

Several algorithms exist to reduce the rank of an ordinary linear differential system at a point, say 0, to its minimal value, the Poincaré rank (also, sometimes called true Poincaré rank). We extend Levelt algorithm, based on the existence of stationary sequences of free lattices, to completely integrable Pfaffian systems with normal crossings in two variables dY = (1/x^p+1 A(x, y)dx + 1/y^q+1B(x, y)dy)Y where A, B are m×m matrices with entries in C[[x, y]] and p, q are non negative integers. The algorithm returns a completely integrable Pfaffian system with normal crossings dZ = (1/x^p+1 A(x, y)dx + 1/y^q+1 B(x, y)dy)Z equivalent to the initial one through a formal meromorphic gauge transformation at the origin 0, the integers p, q being simultaneously and individually the smallest possible. We, thus, set up a first step towards the explicit calculation of formal solutions of such systems.The particular case of a regular singular point at 0 is equivalent to p = q = 0, a condition easily checked by applying the algorithm.

REFERENCES

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

General Terms:
Algorithms

Keywords:
Levelt algorithm, Pfaffian systems, Poincaré rank