On the Series Defined by Differential Equations, With an Extension of the Puiseux Polygon Construction to These Equations

Abstract

In 1903 E.Maillet proved that a formal solution of an algebraic ordinary differential equation has some Gevrey order 8 < oo. In 1989, B.Malgrange gave a bound of s for convergent ordinary differential equations. Here we extend these results for formal differential equations of Gevrey type. Our method is based in an algorithmic use of the Newton-Puiseux Polygon for differential equations.

An extension of the Newton-Puiseux polygon construction to give solutions of Pfaffian forms

“…It is well known that a series solution of a convergent ordinary differential equation has certain Gevrey index (see Maillet [10], Mahler [9], Ramis [14], [15], and Malgrange [11]). In a forthcoming paper [5], we prove that the ring of Gevrey power series is, in some sense, closed for ordinary differential equations. Now we are going to prove that the ring of formal power series with fixed s-Gevrey index has in common with the ring of convergent power series the property of any first order and first degree differential equation with coefficients into the ring has a solution in this ring.…”

“…Actually, it may happen even with the restrictive conditions imposed by Ince [8] that the polynomials <l>(p^)( (7) with /^i > /^o do not have any non zero root, as the following example shows : 5 and let L be the side with slope -1, then <^i) (G) = C(C-1) 2 .…”

The Newton Polygon Method for Differential Equations

“…See [3,4,7,8] for proofs of this theorem in different settings. They are based on the stabilization of the Newton polygon process, which also gives a recurrent formula for the coefficients of φ(x) which we will need later.…”

“…Finally, by Lemma 6 the pivot point of all them has a corresponding monomial of typeḡ x α (xy ), withḡ = 0. This guarantee the convergence of the solutions by a direct application of Theorem 2 of [4] or by the main theorem of [9].…”

“…This section should be read in connection with our main result, stated and proved in section 3, as a technical guide for the proof. All results in section 2 are not original and can be found in [3,4,5,6,7,8]. As an elementary application of the Newton polygon method we have included an algorithm which gives a bound for degree of polynomial solutions of a first order ODE.…”