Oscillating solutions look chaotic. They're algebraically constrained.
Second-order linear ODEs with well-behaved coefficients can produce solutions that oscillate — crossing zero infinitely many times. The oscillation looks irregular: zero spacings vary, amplitudes fluctuate, and the solution has no closed form. The natural assumption is that oscillation is the generic, unstructured case — the solution wiggles because nothing forces it to be simple.
Aschenbrenner, van den Dries, and van der Hoeven (arXiv:2603.02013) resolve a 1982 conjecture of Boshernitzan: oscillating solutions of second-order linear ODEs with Hardy field coefficients decompose into an amplitude function and a phase function whose germs live in a larger Hardy field. Hardy fields are ordered differential fields of germs at infinity — algebraically structured objects closed under differentiation and comparison. The decomposition forces the oscillation into a rigid algebraic framework.
The consequence: the zero set of an oscillating solution is not arbitrary. The spacing between zeros, the growth rate of the amplitude, and the asymptotic behavior of the phase are all constrained by the algebraic properties of the Hardy field extension. Oscillation is structured, not chaotic — the non-oscillation properties of the coefficients propagate into sharp constraints on the oscillatory behavior of solutions.
The conjecture was open for over forty years because the connection between Hardy field arithmetic and oscillation theory required tools from model theory and differential algebra that didn't exist when Boshernitzan posed the question. The answer was always there — the oscillation was always algebraic — but the proof needed the right language.
Aschenbrenner, van den Dries, and van der Hoeven, "Revisiting second-order linear differential equations over Hardy fields," arXiv:2603.02013 (2026).