Genzmer and Mol's study of local polar invariants for dicritical foliation germs (J. Math. Soc. Japan 70 (2018) 1419–1451) explicitly leaves open two questions about the algebraic multiplicity ν0\nu_0 of a singular holomorphic plane foliation. First, given a formal foliation germ with a balanced equation of separatrices F^\hat{F}, the pure algebraic multiplicity ν0∗(F)=ν0(F^)−1\nu_0^*(\mathcal{F})=\nu_0(\hat{F})-1 is defined as the multiplicity that a second-type foliation sharing the same reduction structure and the same balanced equation would have — but the authors cannot guarantee that such a foliation exists: is ν0∗\nu_0^* always realizable as an actual algebraic multiplicity in this sense? Second, their Theorem F states that for topologically equivalent foliation germs having only convergent separatrices, second-typedness and the algebraic multiplicity are topological invariants; whether the convergent-separatrices hypothesis can be dropped — which for the algebraic multiplicity is the two-dimensional case of Mattei's conjecture — remains open. The 2024 work of Câmara–Reis–Sampaio settles several adjacent cases (algebraic multiplicity one, dicritical germs, germs resolved by one blow-up) but leaves the conjecture open. Both questions remain open.

Background

A germ of holomorphic foliation F\mathcal{F} at the origin of C2\mathbb{C}^2 is given by a germ of holomorphic 1-form ω=a(x,y)dx+b(x,y)dy\omega=a(x,y)dx+b(x,y)dy with gcd⁡(a,b)=1\gcd(a,b)=1; its leaves are the integral curves of the associated complex ODE. The algebraic multiplicity ν0(F)\nu_0(\mathcal{F}) is the minimum of the orders at 00 of the coefficients aa and bb — the foliation analogue of the multiplicity of a plane curve singularity. A separatrix of F\mathcal{F} is an irreducible invariant analytic curve, and the Separatrix Theorem of Camacho and Sad guarantees that at least one always exists. Allowing formal power series in the coefficients gives formal foliations and formal (possibly divergent) separatrices, a setting in which the local theory is finer than in the convergent category.

By the reduction of singularities theorem, a finite sequence of point blow-ups turns any foliation germ into finitely many elementary singularities, which are either non-degenerate or saddle-nodes. An exceptional component created by a blow-up is dicritical when the transformed foliation does not preserve it, in which case infinitely many separatrices cross it; the germ is dicritical when its reduction contains such a component. A saddle-node singularity has one strong separatrix, always convergent, and one weak separatrix, which may be divergent; a saddle-node in the final divisor is tangent when its weak separatrix is contained in the divisor. A foliation germ is of second type when no saddle-node in its reduction is tangent — equivalently, when the tangency excess τ(F)\tau(\mathcal{F}), a non-negative integer summing the deviations from transversality at tangent saddle-nodes weighted by the exceptional multiplicities, vanishes — and it is a generalized curve when its reduction contains no saddle-node at all. Non-convergent separatrices arise exactly as the weak separatrices of non-tangent saddle-nodes, while dicritical separatrices always converge; in particular, germs of second type may still possess non-convergent separatrices.

The Poincaré problem asks for a bound on the degree of an invariant algebraic curve SS of a foliation of degree dd on the complex projective plane; in the non-dicritical case Carnicer proved deg⁡(S)≤d+2\deg(S)\le d+2, and classical counterexamples show that no such bound exists in general, the obstruction living in dicritical singularities. To analyze the dicritical case, Genzmer and Mol built local polar invariants on the balanced equation of separatrices: a formal meromorphic function F^\hat{F} whose divisor combines the isolated separatrices and the curvets (separatrices crossing dicritical components) with integer coefficients aD,Ca_{D,C} that, over each dicritical component DD of valence v(D)v(D) (its number of adjacent components), satisfy ∑CaD,C=2−v(D)\sum_C a_{D,C}=2-v(D); in the non-dicritical case this is simply an equation for the finite set of separatrices. The construction, due to Mattei and Salem in the non-dicritical case and to Genzmer in general, satisfies ν0(F)=ν0(F^)−1+τ(F)\nu_0(\mathcal{F})=\nu_0(\hat{F})-1+\tau(\mathcal{F}).

This formula led Genzmer and Mol to define the pure algebraic multiplicity ν0∗(F)=ν0(F^)−1\nu_0^*(\mathcal{F})=\nu_0(\hat{F})-1: the multiplicity with the contribution of tangent saddle-nodes stripped away. It assembles exactly the data of the separatrices and of the reduction structure — the combinatorics of the desingularization tree together with the position of the dicritical components — and would equal the algebraic multiplicity of a second-type foliation sharing that structure and that balanced equation, if such a foliation existed. These pure invariants feed the paper's degree-bound criteria in the dicritical Poincaré problem, which is what makes their interpretation as genuine multiplicities desirable.

The second question belongs to the topological classification program. Two foliation germs are topologically equivalent when a germ of homeomorphism takes leaves of one onto leaves of the other. The Milnor number and the GSV index (a residue-type index of the foliation along a convergent invariant curve) are classical topological invariants, but for the algebraic multiplicity the analogous statement was long known only in classes where all separatrices converge: generalized curves (Camacho, Lins Neto and Sad), non-dicritical second type with convergent separatrices (Mattei and Salem), and — this is Genzmer and Mol's Theorem F — any two topologically equivalent germs all of whose separatrices converge, for which also second-typedness itself is invariant. Rosas proved invariance when the conjugating homeomorphism is differentiable or C1C^1 at the origin. The unrestricted statement — topologically equivalent germs have equal algebraic multiplicity, a foliation analogue of the multiplicity conjecture for hypersurface singularities, proposed by J.-F. Mattei — remains open, and Câmara, Reis and Sampaio (2024) recently settled several new cases: multiplicity one is invariant in dimension two, dicritical plane germs have invariant multiplicity, as do germs resolved by a single blow-up or having, above the first blow-up, a singularity that is locally a strictly non-dicritical second-type germ with convergent separatrices; saddle-node type is also invariant. What no current technique handles is precisely the case motivating the source question: germs possessing non-convergent separatrices, since the index arguments of all existing proofs track only convergent separatrices as anchors.

Problem Statement

The problem comprises the two questions that Genzmer and Mol's paper explicitly leaves open. They share one scientific objective — establishing that the algebraic multiplicity behaves as a genuine multiplicity in the formal, dicritical setting: computable from the balanced equation of separatrices alone, and topologically meaningful. The two questions are independent; a complete answer addresses each, and an answer to either alone is a partial resolution.

Question 1 (realizability of the pure algebraic multiplicity). Let F\mathcal{F} be a germ of formal singular foliation at (C2,0)(\mathbb{C}^2,0) and F^\hat{F} a balanced equation of separatrices for it, so that the pure algebraic multiplicity is ν0∗(F)=ν0(F^)−1\nu_0^*(\mathcal{F})=\nu_0(\hat{F})-1. Does there exist a foliation germ G\mathcal{G} of second type that shares with F\mathcal{F} the same reduction structure — the combinatorics of the desingularization tree together with the position of the dicritical components — and the same balanced equation of separatrices, and whose algebraic multiplicity is ν0(G)=ν0∗(F)\nu_0(\mathcal{G})=\nu_0^*(\mathcal{F})? Prove existence in general, or exhibit a formal foliation germ for whose balanced equation no such second-type foliation exists.

Question 2 (topological invariance without convergent separatrices). Let F1,F2\mathcal{F}_1,\mathcal{F}_2 be germs of holomorphic foliations at (C2,0)(\mathbb{C}^2,0) that are topologically equivalent — there is a germ of homeomorphism taking leaves of F1\mathcal{F}_1 onto leaves of F2\mathcal{F}_2 — with no hypothesis that their separatrices are convergent. Must F1\mathcal{F}_1 be of second type if and only if F2\mathcal{F}_2 is, and must ν0(F1)=ν0(F2)\nu_0(\mathcal{F}_1)=\nu_0(\mathcal{F}_2)? An affirmative answer in general would imply the two-dimensional case of Mattei's conjecture on the equality of algebraic multiplicities, and would subsume the known cases (generalized curves, dicritical germs, algebraic multiplicity one, germs all of whose separatrices converge). A negative answer is an explicit pair of topologically equivalent foliation germs for which one of the two required conclusions fails: either they differ in second-typedness, or they have different algebraic multiplicities.

References

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