Characterizing when the index–Diophantine strategy bounds the degrees of algebraic solutions of non-degenerate polynomial differential equations

For a polynomial differential equation in two variables whose singularities are all non-degenerate — equivalently, a holomorphic foliation of the complex projective plane of degree $n$ defined over an effectively computable field — Coutinho (2018) introduced an algorithmic strategy that computes, from Camacho–Sad-type index sums over the 'singular blocks' of the singular set together with the global identities equating the two total index sums to $m(n+2)$ and $m^2$ , systems of linear Diophantine equations whose non-negative integer solutions constrain the degree $m$ of any algebraic solution (invariant algebraic curve). When the Diophantine systems have finitely many non-negative integer solutions the strategy yields an effective degree bound, but the method is not guaranteed to succeed, and exactly when it does remains open: no non-degenerate foliation with finitely many algebraic solutions is known for which the strategy fails, no criterion decides whether a failure reflects a rational first integral (as in its Example 5.6) or merely a limitation of the bounding method, and the degree-5 search for the foliation of its Example 5.2 was left computationally unfinished. The problem is to characterize the foliations on which the strategy succeeds — in particular, to prove or refute that failure of the Diophantine systems to bound $m$ implies the existence of a rational first integral — and to settle the residual computational regimes, notably the degree-5 case of Example 5.2.

Background

A planar polynomial differential equation can be written as a Pfaffian equation $a(x,y)\,dx+b(x,y)\,dy=0$ , or as the trajectory system of the polynomial vector field $D=b\,\partial_x-a\,\partial_y$ . Homogenizing produces a holomorphic foliation $\mathcal{F}$ of degree $n$ of the complex projective plane $\mathbb{P}^2_{\mathbb{C}}$ — the degree of a foliation is the number of points at which it is tangent to a generic projective line — and its singularities are the common zeros of the homogeneous coefficients of the 1-form defining it. An algebraic solution is an invariant algebraic curve $C=\{h=0\}$ : the polynomial $h$ satisfies $D(h)=g\,h$ for some polynomial $g$ of degree at most $n$ , in which case $h$ is called a Darboux polynomial and $g$ its cofactor. A singularity is non-degenerate when the linear part of the vector field there has two non-zero eigenvalues; their quotient is the characteristic ratio of the singularity. Non-degenerate singularities may still be dicritical — after blowing the point up (replacing it by the projective line of directions through it), the exceptional divisor, the line that replaces the point, is not invariant under the lifted foliation, so separatrices (invariant analytic curves issuing from the singularity) pass through it in infinitely many directions — and dicritical points are precisely where degree bounds become hard.

This is the setting of the classical Poincaré problem: bound the degrees of the algebraic solutions of a foliation in terms of its degree $n$ . The best-known general statement is due to Carnicer: an irreducible invariant curve of degree $m$ that meets no dicritical singularity satisfies $m\le n+2$ . Such uniform bounds, however, require hypotheses, and they are often far from sharp; for actually computing all algebraic solutions of a concrete foliation defined over an effective field (a field, such as $\mathbb{Q}$ , whose elements and ring operations can be represented and computed exactly on a machine), one wants an effective and ideally sharp degree bound, so that a finite undetermined-coefficient search — a Gröbner basis computation, the standard exact method for solving systems of polynomial equations — can enumerate all Darboux polynomials.

The local tool that makes sharper bounds possible is the Camacho–Sad index: at a singularity $p$ lying on an invariant curve $C$ , it is a rational residue computed from the eigenvalue data of the linear part, and the Camacho–Sad theorem states that the sum of these indices over all singularities of $\mathcal{F}$ on $C$ equals the curve's self-intersection number $C\cdot C=m^2$ . A companion index (the quotient of the eigenvalues in the transverse direction) sums to $m(n+2)$ , the degree of the restriction to $C$ of the foliation's normal bundle (the line bundle of directions transverse to the leaves), which has degree $n+2$ . Writing $\iota_1$ and $\iota_2$ for these two total sums, the global identities are $\iota_1=m(n+2)$ and $\iota_2=m^2$ (Theorem 2.6 of Coutinho (2018)). Coutinho (2018) turns these two global identities into an algorithm: after placing $\mathcal{F}$ in generic $x$ -position — a coordinate change after which the singular set decomposes as a disjoint union of 'singular blocks' $S=\{(a,g_S(a)):f_S(a)=0\}$ with $f_S$ square-free, computed by a factorizing Gröbner-basis algorithm that exploits the Shape Lemma (the standard fact that a zero-dimensional radical ideal in generic coordinates admits a Gröbner basis consisting of one univariate polynomial in the last variable together with polynomials of degree one in the others) — Galois conjugacy ties the characteristic ratios of the points of a block together, so the two block-wise index sums are either explicit rational numbers — when no characteristic ratio in the block is a positive rational — or linear forms in non-negative integer parameters attached to the block. Substituting one admissible pair of block sums per block into the two global identities produces pairs of linear Diophantine equations in the unknown curve degree $m$ and the block parameters. Whenever these systems have altogether finitely many non-negative integer solutions, the degrees of all algebraic solutions are bounded (and the surviving solutions carry multiplicity information that prunes the subsequent search); the systems are solved by classical linear-Diophantine algorithms after an $LDL^t$ diagonalization of the associated quadratic form, which factors the symmetric matrix of the form into lower-triangular, diagonal, and transpose factors and separates the variables bounded by the positive-definite diagonal part from those that remain free.

The paper implements the procedure in Axiom over $\mathbb{Q}$ and works out six examples. For the degree-3 foliation $\mathcal{F}_2$ of its Example 5.2 the systems force $m\in\{2,3,5\}$ , node-counting constraints (from the classical genus formula for singular plane curves) exclude $m=2$ , and an undetermined-coefficient search finds one irreducible cubic solution, $h=2y^3+(6x+1)y^2+(6x^2+4x)y+2x^3+2x^2$ ; whether degree-5 solutions exist was left open, the corresponding coefficient system being too large for a standard Gröbner computation. For $\mathcal{F}_6$ (Example 5.6) the systems fail to bound $m$ — after diagonalization, the linear equation for $m$ involves integer parameters not bounded by the positive-definite part of the quadratic one — and $\mathcal{F}_6$ indeed has a rational first integral, exhibited through four quadratic Darboux polynomials whose cofactors satisfy an exact linear relation. Classical Darboux theory provides the relevant dichotomy: a foliation of the plane has infinitely many algebraic solutions exactly when it admits a rational first integral, i.e. a non-constant rational function constant along trajectories, built as a quotient of products of Darboux polynomials. Thus the failure of the method on $\mathcal{F}_6$ is 'explained' by genuine integrability; the open question is whether this is the only possible failure mode.

Problem Statement

Let $\mathcal{F}$ be a foliation of $\mathbb{P}^2_{\mathbb{C}}$ of degree $n$ , defined over an effective field $K$ (for instance $\mathbb{Q}$ ), all of whose singularities are non-degenerate. Apply the procedure of Coutinho (2018): put $\mathcal{F}$ in generic $x$ -position; decompose its singular set into singular blocks $S$ ; compute, for each block, the possible values of the two block-wise index sums — explicit rationals when no characteristic ratio of the block is a positive rational, and linear forms in non-negative integer parameters otherwise; and substitute one admissible pair per block into the global identities $\iota_1=m(n+2)$ and $\iota_2=m^2$ — where $\iota_1$ and $\iota_2$ denote the total sums, over all singularities lying on a candidate invariant curve of degree $m$ , of the companion index (the eigenvalue quotient in the transverse direction) and the Camacho–Sad index respectively — to obtain systems of linear Diophantine equations in the curve degree $m$ and the block parameters. Say the strategy succeeds on $\mathcal{F}$ when these systems have altogether finitely many non-negative integer solutions, so that the degrees of all algebraic solutions of $\mathcal{F}$ are effectively bounded.

(1) Characterize the class of non-degenerate foliations on which the strategy succeeds. The central question is to prove or refute that if the strategy fails on $\mathcal{F}$ — the Diophantine systems admit infinitely many non-negative integer solutions — then $\mathcal{F}$ has a rational first integral, equivalently infinitely many algebraic solutions. A refutation must exhibit an explicit $\mathcal{F}$ over an effective field (for instance a number field given by explicit coefficients), with only finitely many algebraic solutions (possibly none), whose systems fail to bound $m$ ; no such example is currently known, and settling whether one exists is part of the question.

(2) Settle the residual computational regime that the method leaves open: for the degree-3 foliation $\mathcal{F}_2$ of Example 5.2 of Coutinho (2018) — defined there by explicit rational coefficients, with algebraic-solution degrees constrained to $\{2,3,5\}$ , degree $2$ excluded, and one irreducible cubic solution known — decide whether degree-5 algebraic solutions exist.

The common objective of (1) and (2) is to convert the index–Diophantine strategy from a heuristic into a method of proven scope: to decide, within the class of non-degenerate foliations over effective fields, exactly when singularity data suffice to bound the degrees of algebraic solutions, and to make the residual cases effectively decidable.

References

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