Solving ordinary differential equations in terms of series with real exponents

Abstract:

We generalize the Newton polygon procedure for algebraic equations to generate solutions of polynomial differential equations of the form $\sum \nolimits _{i = 0}^\infty {{\alpha _i}{x^{{\beta _i}}}}$ where the ${\alpha _i}$ are complex numbers and the ${\beta _i}$ are real numbers with ${\beta _0} > {\beta _1} > \cdots$. Using the differential version of the Newton polygon process, we show that any such a series solution is finitely determined and show how one can enumerate all such solutions of a given polynomial differential equation. We also show that the question of deciding if a system of polynomial differential equations has such a power series solution is undecidable.