Formal Solutions of Linear PDEs and Convex Polyhedra

Abstract

The Newton polygon construction for ODEs, and Malgrange-Ramis polygon for partial differential equations in one variable are generalized in order to give an algorithm to find solutions of a linear partial differential equation at a singularity. The solutions found involve exponentials, logarithms and Laurent power series with exponents contained in a strongly convex cone.

Rank reduction of a class of pfaffian systems in two variables

“…It remains to show thatp is equal to the Poincaré rank of Δ (1) . By construction, the columns of Pj form an O-basis βj of Λ (j) and hence an O (1) -basis of Λ (j) (1) .…”

“…, pn) is called the rank of (1). This paper deals with the following rank reduction problem: Given a system (1), to find a formal meromorphic gauge transformation Y = T (x)Z, T ∈ GLm(K), that takes (1) into an equivalent completely integrable Pfaffian system with normal crossings and minimal rank 8 > > < > > :…”

Puiseux parametric equations of analytic sets

“…6], as solutions of partial differential equations [4], and as solutions of holonomic systems [16]. Also, A. D. Bruno uses them in [6] to construct a method for solving non-linear differential equations.…”

Some geometric aspects of Puiseux surfaces

“…In the quasi-ordinary case, this is the celebrated Jung-Abhyankar theorem ( [5], [1]) but, for the more general case, we cannot know whether a given polynomial has Puiseux roots unless we actually compute them. This is now posible, thanks to the work of McDonald ( [9]) and Aroca-Cano ( [2]); but, while useful for working with examples, this method has not been adopted so far for its use on abstract argumentations.…”