A Reduction for Regular Differential Systems

We propose a definition of regularity of a linear differential system with coefficients in a monomial extension of a differential field, as well as a global and truly rational (i.e. factorisation–free) iteration that transforms a system with regular finite singularities into an equivalent one with simple finite poles. We then apply our iteration to systems satisfied by bases of algebraic function fields, obtaining algorithms for computing the number of irreducible components and the genus of algebraic curves. Introduction This paper is concerned with differential systems of the form  y ′ 1 .. y′ n  = A  y1 .. yn  (1) where the entries of the matrix A are in a differential field F . While the cyclic vector method [2, 5, 14] reduces (in theory) the study of such systems to the study of scalar differential equations, that method is well-known to be impractical except for very small n (see for example the timings in [2]), which motivates the study of direct algorithms that do not require uncoupling the system (1). Direct algorithms are based on computing T ∈ GLn(F ) such that the change of variable Y = TZ transforms (1) into an equivalent differential system Z ′ = BZ where the matrix B has some desired property, which can be related either to its shape (e.g. triangular) or the nature of its poles. For example, when F is the rational function field C(x) with the derivation d/dx, the Moser form algorithm [12] yields a matrix B whose denominator b(x) divides the denominator of A, and such that the multiplicity of x as a factor of b(x) is minimal among all T ∈ GLn(C(x)). Knowledge of that minimal multiplicity determines