Open problems completing the zero-dimensional infinitesimal and tangential 16th Hilbert problem: higher-order Melnikov functions, Bautin index, bifurcation diagrams, real zero counts, and chains

The zero-dimensional analogue of the infinitesimal and tangential 16th Hilbert problems asks to count, in terms of the degrees $m$ and $n$ of two univariate polynomials $f$ and $g$, the isolated zeros of the displacement function $\Delta(t,\epsilon)$ of the deformation $f+\epsilon g$ along a zero-cycle of $f$, and of its first-order term, the Abelian integral $M_1(t)$. Bravo, Mardešić, Novikov and Pontigo-Herrera (2023 preprint, published 2025) solved both problems completely with optimal bounds, exhibited 'alien' limit cycles invisible at first order, and disproved the asymptotic Chebyshev conjecture for zero-cycles. Their concluding section poses a precise list of problems that complete the theory beyond the solved first-order, complex, cycle setting: (a) the structure and zero counts of higher-order Melnikov functions $M_\mu$, including an explicit degree-6 example with $M_1\equiv0$ and $M_2\not\equiv0$; (b) bifurcation diagrams of $\Delta$ and $M_1$ in the parameter space of perturbations for low degrees; (c) a local Bautin problem near critical values where the cycle degenerates; (d) explicit bounds on the Bautin index $b(m)$; (e) zero counts and the possible Chebyshev property on real intervals between critical values; and (f) extension from cycles to chains. No resolution of any of these questions is known in the literature; the related Noetherianity theorem for Melnikov functions of planar foliations (2025/2026) concerns a different, planar setting. The problem is open.

Background

The classical infinitesimal 16th Hilbert problem concerns limit cycles born when an integrable planar polynomial vector field is slightly perturbed. Writing the Poincaré return map — the map sending a point of a closed oval of the unperturbed flow to its position after one revolution of the perturbed flow — along the ovals of a Hamiltonian first integral as a displacement expansion $\Delta(t,\epsilon)=\sum_{j\ge1}M_j(t)\epsilon^j$, the first-order term $M_1(t)$ is an Abelian integral, i.e. an integral of a polynomial one-form along a one-dimensional cycle of a level curve; counting its isolated zeros is the tangential problem, and counting the zeros of the full displacement is the infinitesimal problem. In that planar (1-cycle) setting, the only general result is a very large explicit bound for the tangential problem (Binyamini, Novikov, Yakovenko, 2010), and no general bound is known for the infinitesimal problem.

Bravo, Mardešić, Novikov and Pontigo-Herrera (2025) study the zero-dimensional analogue, which becomes purely algebraic. Let $f,g$ be polynomials in one complex variable with $\deg f=m>1$, $\deg g=n$; let $\Sigma$ be the finite set of critical values of $f$ (images of the zeros of $f'$). For a regular value $t\notin\Sigma$, the fiber $f^{-1}(t)$ consists of $m$ points $z_1(t),\dots,z_m(t)$, analytic multivalued functions of $t$ on $\mathbb{C}\setminus\Sigma$. A zero-dimensional chain of $f$ is a formal integer combination $C(t)=\sum_{j=1}^m n_j z_j(t)$; it is a zero-dimensional cycle when $\sum_j n_j=0$, and a simple cycle when $C=z_2-z_1$. By convention, the 'integral' of a function $h$ along such a chain is the finite evaluation sum $\int_{C(t)}h=\sum_j n_j h(z_j(t))$. Under the deformation $f\mapsto f+\epsilon g$ each preimage moves to a solution $z_j(t,\epsilon)$ of $f+\epsilon g$, and the displacement function of the cycle is

$$\Delta(t,\epsilon)=\int_{C_\epsilon(t)}f=-\epsilon\int_{C(t)}g+o(\epsilon)=\epsilon M_1(t)+o(\epsilon),$$

where the first Melnikov function $M_1(t)=-\int_{C(t)}g$ is an Abelian integral over the zero-cycle. Unlike the planar case, $M_1$ and $\Delta$ are algebraic multivalued functions with finitely many determinations, and zeros are counted at regular values over all determinations; for $\Delta$ one counts the supremum over sufficiently small $\epsilon$ (a cyclicity-type count). The tangential (respectively infinitesimal) 16th Hilbert problem on zero-cycles asks for the maximal numbers $Z_1(m,n)$ (respectively $Z_\Delta(m,n)$) of isolated zeros, over all $f,g$ of degrees at most $m,n$ and all cycles of $f$.

Prior work: Gavrilov and Movasati (2007) studied the tangential problem on simple zero-cycles and obtained the upper bound $(m-1)(n-1)/2$; they conjectured an asymptotic Chebyshev property, i.e. that the ratio of the maximal zero count to the dimension of the space of Abelian integrals tends to 1. The related center problems on zero-cycles (deciding when all Melnikov functions vanish identically) were solved inductively by Álvarez, Bravo and Mardešić (2013, tangential case) and by Álvarez, Bravo, Christopher and Mardešić (2021, infinitesimal case). Bravo, Mardešić, Novikov and Pontigo-Herrera (2025) completed the theory: it determined $Z_1(m,n)$ and $Z_\Delta(m,n)$ exactly for all $m,n$ (for instance $Z_1(m,n)=n(m-1)!$ when $m>2$ and $m$ does not divide $n$); showed that the tangential and infinitesimal problems are not equivalent, determining the degree configurations in which 'alien' limit cycles — zeros of $\Delta$ with no first-order counterpart — can occur for generic data; and proved that the asymptotic Chebyshev conjecture fails very strongly: $Z_1(m,m-1)/\dim\mathcal{P}(m,m-1)\to\infty$ even for simple cycles, where $\mathcal{P}(m,n)$ denotes the space of Abelian integrals in question. The method is geometric: zeros of $M_1$ are read off from the intersection of the connection curve $\Gamma_f\subset\mathbb{CP}^m$ (a curve in complex projective $m$-space recording which branches of the multivalued inverse of $f$ are connected through the cycle) with the zero hypersurface $S_g$ of the perturbation. A finite-dimensional function space is called a Chebyshev system if each of its nonzero elements has fewer zeros than the dimension of the space; a Bautin-type index measures how many Taylor coefficients of a displacement must vanish before all of them do. The following questions are open.

Problem Statement

The source paper solved the tangential and infinitesimal 16th Hilbert problems on zero-cycles by determining the optimal values of $Z_1(m,n)$ and $Z_\Delta(m,n)$ for all degrees. Complete the remaining quantitative theory of the displacement expansion $\Delta(t,\epsilon)=\sum_{j\ge1}M_j(t)\epsilon^j$ of the zero-cycle deformation $f+\epsilon g$ by resolving the following problems posed in the paper's concluding section (zeros are counted at regular values of $f$, over all determinations of the multivalued functions involved, and for $\Delta$ in the cyclicity sense over small $\epsilon$). Each of (a)-(f) is posed as a separate open question.

(a) Higher-order Melnikov functions. The terms $M_j$ of the displacement expansion are called Melnikov functions. When $M_1\equiv0$, the first nonzero Melnikov function $M_\mu$ is an Abelian integral in a generalized sense: $M_\mu(t)=\int_{C(t)}P(z)/(f'(z))^\mu$ with $P$ a polynomial (in the zero-cycle evaluation convention above). For fixed degree bounds on $f$ and $g$, determine the structure of the space of first nonzero Melnikov functions $M_\mu$ and bound the number of their isolated zeros. In addition to the general structure above, study the explicit example $f=z^6$, $C(\epsilon)=z_0+2z_1+z_2-z_3-2z_4-z_5$, $g=z^2+z^3$, for which $M_1\equiv0$ and $M_2\not\equiv0$.

(b) Bifurcation diagrams. Describe the bifurcation diagram of the displacement function $\Delta$ or of its tangential part $M_1$ in the parameter space of polynomials $g$ for low degrees, i.e. for degrees $(m,n)$ with $2\le m\le3$ and $2\le n\le3$: the bifurcation diagram — the set where the number, location, or multiplicity of regular zeros changes — is an algebraic set over $\mathbb{C}$ and a semi-algebraic set (defined by polynomial equalities and inequalities) over $\mathbb{R}$; characterize this stratification explicitly.

(c) Local Bautin problem near a degenerate cycle. For a deformation $f+\epsilon g$ of degree at most $m$, a cycle $C(t)$ of $f$, and a critical value $t_0$ of $f$ at which $C(t)$ degenerates, bound, in function of $m$, the number of regular solutions of $\Delta(t,\epsilon)=0$ in a neighborhood of $t_0$, and determine the maximal multiplicity of such cycles as a function of the degrees $m$ and $n$.

(d) Bautin index. For $f$ and $g$ of degree at most $m$ and any cycle $C(t)$, the coefficients $M_j$ of the displacement expansion generate an ascending chain of ideals $(M_1)\subset(M_1,M_2)\subset\cdots$ in the polynomial ring of the coefficients of the perturbation $g$, which stabilizes at some index $b(m)$ by Noetherianity. Bound the Bautin index $b(m)$ explicitly as a function of $m$, uniformly over $f$ and $C$.

(e) Real restriction. For real polynomials $f,g$ and $t$ ranging over an interval between two adjacent real critical values of $f$, bound the number of real zeros of the real determinations of $M_1$ or of $\Delta(t,\epsilon)$, and decide whether a Chebyshev property (fewer zeros than the dimension of the underlying function space) can hold in this real setting.

(f) Chains. Generalize the previous results from cycles to arbitrary chains, i.e. drop the condition $\sum_j n_j=0$ on the coefficients, and determine which parts of the theory extend and how the counting formulae change.

The paper also poses a further, deliberately motivational question — extending the connection-curve and zero-hypersurface technique to the generalized zero-cycle problems with multivalued integrands obtained by reducing the tangential and infinitesimal problems on one-dimensional cycles (planar Hamiltonian systems) to zero-cycles, with the aim of attacking those classical problems. That question points toward the classical infinitesimal and tangential 16th Hilbert problems for planar vector fields, which are famous standalone open problems; it is not part of the required scope of this problem.

References

1. J. L. Bravo, P. Mardešić, D. Novikov, J. Pontigo-Herrera, "Infinitesimal and tangential 16-th Hilbert problem on zero-cycles", Bulletin des Sciences Mathématiques 202 (2025), article 103634. DOI: 10.1016/j.bulsci.2025.103634; preprint arXiv:2312.03081. https://doi.org/10.1016/j.bulsci.2025.103634
2. L. Gavrilov, H. Movasati, "The infinitesimal 16th Hilbert problem in dimension zero", Bulletin des Sciences Mathématiques 131 (2007), no. 3, 242-257. DOI: 10.1016/j.bulsci.2006.04.003. https://doi.org/10.1016/j.bulsci.2006.04.003
3. A. Álvarez, J. L. Bravo, P. Mardešić, "Vanishing Abelian integrals on zero-dimensional cycles", Proceedings of the London Mathematical Society (3) 107 (2013), no. 6, 1302-1330. DOI: 10.1112/plms/pdt012. https://doi.org/10.1112/plms/pdt012
4. A. Álvarez, J. L. Bravo, P. Mardešić, "Inductive Solution of the Tangential Center Problem on Zero-Cycles", Moscow Mathematical Journal 13 (2013), no. 4, 555-583. DOI: 10.17323/1609-4514-2013-13-4-555-583. https://doi.org/10.17323/1609-4514-2013-13-4-555-583
5. A. Álvarez, J. L. Bravo, C. Christopher, P. Mardešić, "Infinitesimal Center Problem on Zero Cycles and the Composition Conjecture", Functional Analysis and Its Applications 55 (2021), no. 4, 257-271. DOI: 10.1134/S0016266321040018; preprint arXiv:2006.07600. https://doi.org/10.1134/S0016266321040018
6. G. Binyamini, D. Novikov, S. Yakovenko, "On the number of zeros of Abelian integrals", Inventiones mathematicae 181 (2010), no. 2, 227-289. DOI: 10.1007/s00222-010-0244-0. https://doi.org/10.1007/s00222-010-0244-0
7. L. Gavrilov, "Higher order Poincaré-Pontryagin functions and iterated path integrals", Annales de la Faculté des sciences de Toulouse: Mathématiques (6) 14 (2005), no. 4, 663-682. DOI: 10.5802/afst.1107. https://doi.org/10.5802/afst.1107
8. P. Mardešić, D. Novikov, L. Ortiz-Bobadilla, J. Pontigo-Herrera, "Noetherianity and length of Melnikov functions", Bulletin of the Brazilian Mathematical Society, New Series 57 (2026), no. 3, article 34. DOI: 10.1007/s00574-026-00521-7; preprint arXiv:2512.20045. https://doi.org/10.1007/s00574-026-00521-7