The algebraic-critical case of the Poincaré problem: which local algebraic solutions extend to global invariant algebraic curves

For a planar polynomial differential equation, the Poincaré problem asks for an effective bound on the degree of its invariant algebraic curves (Darboux polynomials). Lei and Yang reduced such bounds, for systems admitting an invariant straight line, to the algebraic multiplicities of the singular points on that line — the numbers of distinct formal Puiseux-series solutions at those points — but their bounds become vacuous whenever a singularity is algebraic-critical, meaning it carries infinitely many such formal local algebraic solutions. They posed, and left open, the local-to-global question: at an algebraic-critical singular point, determine which of the infinitely many formal local algebraic solutions actually extend to branches of global algebraic solutions (irreducible invariant algebraic curves), with the aim of obtaining computable degree bounds for systems with algebraic-critical singularities — a step they expected to lead toward a final resolution of the Poincaré problem. Two decades of adjacent literature include Carnicer's non-dicritical bound, Galindo and Monserrat's solution for foliations with a single dicritical divisor, Genzmer and Mol's balanced-equation inequality with conditional bounds, Demina's Puiseux-series degree bounds, and Cano, Fortuny Ayuso and Ribón's local Poincaré problem. The question remains unresolved: no effective criterion for the extendability of local algebraic solutions at algebraic-critical points, and no computable degree bound covering such points in general, is known.

Background

A planar polynomial system is a vector field $\dot z=P(z,w)$, $\dot w=Q(z,w)$ on the complex projective plane $\mathbb{P}^2_{\mathbb{C}}$, where $P$ and $Q$ are relatively prime complex polynomials. A polynomial $f(z,w)$ is a Darboux polynomial (equivalently, its zero set is an invariant algebraic curve) if $P(z,w)\,\frac{\partial f}{\partial z}+Q(z,w)\,\frac{\partial f}{\partial w}=R_f(z,w)\,f(z,w)$ for some polynomial cofactor $R_f$. Invariant algebraic curves are the basic building blocks of Darboux integrability theory: sufficiently many of them force a first integral (a non-constant function constant along trajectories), and they are the level sets of rational first integrals. The Poincaré problem, which goes back to Poincaré, asks for an upper bound on the degree of the invariant algebraic curves of such a system — originally as the key step in deciding algebraic integrability by finite computation. A uniform bound depending only on the degree of the system does not exist: classical families such as the degree-one foliations with homogeneous first integrals $x^p z^{q-p}/y^q$ (whose generic invariant fiber has degree $q$ with $q$ arbitrary) and the degree-four pencil of Lins Neto (a one-parameter family of degree-four foliations) admitting rational first integrals of unbounded degree, both discussed by Genzmer and Mol, show that no such bound is possible. For any fixed system a bound does exist, but no effective way to compute one from the coefficients is known in general, and algorithms that compute rational first integrals and Darboux polynomials must currently take the degree bound as an external input. The strongest general positive result covers the non-dicritical case. A singular point of the induced holomorphic foliation (the partition of the projective plane into the vector field's complex integral curves) is called dicritical if infinitely many local invariant curves (separatrices) pass through it, and non-dicritical otherwise. Carnicer proved that if an invariant algebraic curve $C$ contains no dicritical singularities, then $\deg C\le\deg\mathcal{F}+2$; Cerveau and Lins Neto had earlier proved the same bound when all singularities on $C$ are nodal, and Walcher gave an elementary proof of a $M+1$ bound under non-degeneracy conditions at the stationary points at infinity. All of these hypotheses exclude precisely the dicritical singularities. Lei and Yang proposed a different, computationally oriented approach through local algebraic solutions. Writing the system (after a linear change placing an invariant line at $z=0$) as $dw/dz=P(z,w)/(z\,Q(z,w))$ of degree $M=\max\{\deg P,\deg z\,Q\}$, a local algebraic solution at a point $(z_0,w_0)$ is a non-constant formal Puiseux series $w(z)=w_0+\sum_{i\ge0}\alpha_i(z-z_0)^{\mu_i}$ with $\mu_i\in\mathbb{Q}_+$ strictly increasing that formally solves the equation. The algebraic multiplicity $\mathrm{Mul}(z_0,w_0)$ is the number of distinct such series; a point is called algebraic-critical when this number is infinite. A Newton-polygon criterion (their Lemma 2.2; the Newton polygon is the convex hull, in the exponent plane, of the monomials of the differential equation) gives an effective sufficient condition for algebraic criticality: if a side of the Newton polygon satisfies a rationality condition, a whole one-parameter family $w(z)=z^{\lambda}u(z^{1/q};\alpha)$ of formal algebraic solutions exists. Algebraic-critical points are dicritical, but the converse fails (points can be dicritical without being algebraic-critical), so the non-algebraic-critical hypothesis is genuinely weaker than Carnicer's non-dicritical hypothesis. Their main theorems state: if $a_1,\dots,a_k$ are the roots of $P(0,w)=0$ (plus $a_0=\infty$), then any irreducible strict Darboux polynomial (strict meaning that its zero set contains no line component $z=z_0$ or $w=w_0$) satisfies $\deg_w f\le\sum_{i=0}^{k}\mathrm{Mul}(0,a_i)$, and in particular $\deg_w f\le M(k+1)$ when no $(0,a_i)$ is algebraic-critical; and a system of degree $M$ with an invariant straight line whose singular points are not algebraic-critical satisfies $\deg f\le M(M+1)$. At an algebraic-critical point the multiplicity is infinite, so the counting bound degenerates. Yet the infinitely many local formal solutions come in one-parameter families, and it is not automatic that infinitely many of them are branches of actual global algebraic solutions; deciding which local branches glue to global algebraic factors is exactly the gap between the local Newton-polygon computation and a global degree bound.

Problem Statement

Consider a polynomial system $\dot z=P(z,w)$, $\dot w=Q(z,w)$ in the complex projective plane, with $P,Q$ relatively prime, and let a singular point of the associated differential equation $dw/dz=P(z,w)/(z\,Q(z,w))$ (the form used by Lei and Yang, obtained after a linear change of variables placing an invariant line at $z=0$; its degree is $M=\max\{\deg P,\deg z\,Q\}$) be algebraic-critical in the sense of Lei and Yang: the number of distinct non-constant formal Puiseux-series solutions $w(z)$ at that point is infinite (equivalently, the algebraic multiplicity $\mathrm{Mul}$ is infinite; an effective sufficient condition is given by their Newton-polygon criterion, Lemma 2.2). The problem, stated at the end of Section 3 of the source, is the local-to-global question:

(1) Determine which of the infinitely many formal local algebraic solutions at an algebraic-critical singular point extend to global algebraic solutions — that is, are branches of algebraic functions defined by irreducible Darboux polynomials of the system (a Darboux polynomial is a polynomial $f$ satisfying $P\,\partial f/\partial z+Q\,\partial f/\partial w=R_f f$ for some polynomial cofactor $R_f$; its zero set is an invariant algebraic curve) — and thereby render the contribution of an algebraic-critical point to the degree count finite and effectively computable. The count may legitimately be infinite for systems with a rational first integral, so the target is a characterization (in terms of the coefficients, the reduction of singularities — the resolution of a singular point into elementary ones by a finite sequence of point blow-ups — or equivalent effective data) deciding extendability, not a universal finite number.

(2) As the derived objective that the source itself attaches to this question, use the answer to extend the explicit computable degree bounds of Lei and Yang — $\deg_w f\le\sum_{i=0}^{k}\mathrm{Mul}(0,a_i)$, where $a_1,\dots,a_k$ are the finite roots of $P(0,w)=0$ on the invariant line and $a_0=\infty$, together with its corollaries $\deg_w f\le M(k+1)$ and $\deg f\le M(M+1)$ for systems with an invariant straight line whose singular points are not algebraic-critical — to polynomial systems that have algebraic-critical singularities on the invariant line, or to characterize the obstructions that still permit a finite global degree bound despite local algebraic-critical behavior, moving toward an effective resolution of the Poincaré problem in its remaining (dicritical-type) case.

Parts (1) and (2) are one coherent objective: an effective local-to-global theory for invariant algebraic curves at algebraic-critical singularities, with the degree bounds as the objective the source itself attaches to the question.

References

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