Formal power series solutions of the differential equation ay'=by^m over Dedekind domains of characteristic zero

Abstract

Let D be a Dedekind domain of characteristic zero, let K be the fraction field of D, and consider the Cauchy problem ay′ = byᵐ, y(0) = c₀, in the ring D[[x]], where a, b, c₀ ∈ D, a, b ≠ 0, and m ∈ ℕ. The paper studies when the unique formal solution y ∈ K[[x]] with initial value c₀ actually has all coefficients in D, and therefore belongs to D[[x]]. The argument starts from the coefficient recursion in K[[x]] and derives an explicit formula for the coefficients of the unique solution. This reduces the existence problem in D[[x]] to an arithmetic integrality question governed by the valuations attached to nonzero prime ideals of D. The main point is that over a Dedekind domain the relevant obstruction is not ordinary divisibility by prime elements, but comparison of valuations of fractional ideals after localization at prime ideals.

For the linear case m = 1, the zero initial value always gives the zero solution. For c₀ ≠ 0, the coefficients contain factorial denominators, and this produces a global obstruction unless only finitely many rational primes remain nonunits in D. In that finite-prime situation the exact criterion for a nonzero initial value is the ideal containment (b) ⊆ (a)r₁, where r₁ is an explicitly described correction ideal built from the prime ideals lying over the rational primes that are nonunits in D. For the nonlinear case m ≥ 2, writing d = m − 1, the coefficients are expressed through the product Cₖ(d) = ∏(di + 1), where i runs from 0 to k − 1. The paper shows that prime ideals over rational primes not dividing d create no additional obstruction beyond the basic condition (bc₀ᵈ) ⊆ (a), while the prime ideals over rational primes dividing d contribute a correction ideal r(d). As a result, the exact existence criterion becomes (bc₀ᵈ) ⊆ (a)r(d). In particular, for m = 2 one has r(1) = D, so the condition reduces to (bc₀) ⊆ (a). Several examples are included to illustrate the theorem over ℤ, localizations of ℤ, Gaussian integers, and the Dedekind domain ℤ[√−5], which is not a unique factorization domain.