Scalar Polynomial Vector Fields in Real and Complex Time

Recent PDE studies address global boundedness versus finite-time blow-up in equations like the quadratic parabolic heat equation versus the nonconservative quadratic Schrödinger equation. The two equations are related by passage from real to purely imaginary time. Renewed interest in pioneering work by Masuda, in particular, has further explored the option to circumnavigate blow-up in real time, by a detour in complex time.
In the present paper, the simplest scalar ODE case is studied for polynomials
\begin{equation*} \dot{w}=f(w)=(w-e_0)\cdot\ldots\cdot(w-e_{d-1}), \tag{*} \end{equation*}
of degree $d$ with $d$ simple complex zeros. The explicit solution by separation of variables and explicit integration is an almost trivial matter.
In a classical spirit, indeed, we describe the complex Riemann surface $\mathcal{R}$ of the global nontrivial solution $(w(t),t)$ in complex time, as an unbranched cover of the punctured Riemann sphere $w\in\widehat{\mathbb{C}}_d:=\widehat{\mathbb{C}}\setminus\{e_0,\ldots,e_{d-1}\}$. The flow property, however, fails at $w=\infty\in\widehat{\mathbb{C}}_d$. The global consequences depend on the period map of the residues $2\pi\mathrm{i}/f'(e_j)$ of $1/f$ at the punctures, in detail. We therefore show that polynomials $f$ exist for arbitrarily prescribed residues with zero sum. This result is not covered by standard interpolation theory.
Motivated by the PDE case, we also classify the planar real-time phase portraits of (*). Here we prefer a Poincaré compactification of $w\in\mathbb{C}=\mathbb{R}^2$ by the closed unit disk. This regularizes $w=\infty$ by $2(d-1)$ equilibria, alternately stable and unstable within the invariant circle boundary at infinity. In structurally stable hyperbolic cases of nonvanishing real parts $\Re f'(e_j)\neq 0$, for the linearizations at all equilibria $e_j$, and in the absence of saddle-saddle heteroclinic orbits, we classify all compactified phase portraits, up to orientation-preserving orbit equivalence and time reversal. Combinatorially, their source/sink connection graphs correspond to the planar trees of $d$ vertices or, dually, the circle diagrams with $d-1$ nonintersecting chords. The correspondence provides an explicit count of the above equivalence classes of ODE (*), in real time.
We conclude with a discussion of some higher-dimensional problems. Not least, we offer a 1,000 € reward for the discovery, or refutation, of complex entire homoclinic orbits.