A real planar polynomial vector field of degree mm can have limit cycles that are ovals of invariant algebraic curves (algebraic limit cycles). Llibre, Ramírez and Sadovskaia posed the question of whether the number of algebraic limit cycles is bounded in terms of mm alone, and conjectured that the maximum is 1+(m−1)(m−2)/21+(m-1)(m-2)/2. Zhang proved this conjectured value (in its parity-refined form, 1+(m−1)(m−2)/21+(m-1)(m-2)/2 for even mm and (m−1)(m−2)/2(m-1)(m-2)/2 for odd mm) for fields whose invariant algebraic curves are all nodal, and answered the boundedness question for non-dicritical invariant curves with a weaker bound, which he conjectured to be non-optimal. The general conjecture remains open — even for quadratic systems — together with the two explicitly posed open sub-questions: whether the non-dicritical bound can be improved to the nodal value, and whether a degree-mm holomorphic singular foliation of CP(2)\mathbb{CP}(2) can carry a non-dicritical irreducible invariant algebraic curve of degree m+1m+1 or m+2m+2.

Background

The second part of Hilbert's 16th problem asks for an upper bound, depending only on the degree mm, on the number of limit cycles of a real planar polynomial differential system x˙=p(x,y),y˙=q(x,y)\dot x=p(x,y), \dot y=q(x,y) with max⁡(deg⁡p,deg⁡q)=m\max(\deg p,\deg q)=m; it remains one of the central open problems of the qualitative theory of ordinary differential equations. A limit cycle is a periodic orbit isolated in the set of all periodic orbits. An algebraic curve f=0f=0 with f∈C[x,y]f\in\mathbb{C}[x,y] is an invariant algebraic curve of the vector field X=p∂x+q∂yX=p\partial_x+q\partial_y if Xf=pfx+qfy=KfXf=p f_x+q f_y=Kf for some polynomial KK, called the cofactor. An algebraic limit cycle is a limit cycle contained in an invariant algebraic curve (a closed oval of such a curve). Because the Darboux theory of integrability — which builds first integrals and integrating factors of a vector field from products of powers of its invariant algebraic curves — depends on these curves, Llibre, Ramírez and Sadovskaia (2010) singled out a tractable 'simple version' of Hilbert's 16th problem: is there an upper bound, depending only on mm, on the number of algebraic limit cycles of all polynomial vector fields of degree mm? They solved it for fields whose irreducible invariant algebraic curves are generic (non-singular, meeting pairwise transversally, at most two through any point, and with no common factors at infinity), where the maximum is 1+(m−1)(m−2)/21+(m-1)(m-2)/2 for even mm and (m−1)(m−2)/2(m-1)(m-2)/2 for odd mm — the same parity-dependent values as Harnack's classical bound on the number of ovals of a real plane curve of degree nn, namely 1+(n−1)(n−2)/21+(n-1)(n-2)/2 ovals for even nn and (n−1)(n−2)/2(n-1)(n-2)/2 for odd nn — and they conjectured 1+(m−1)(m−2)/21+(m-1)(m-2)/2 as the universal maximum. A parallel degree bound comes from foliation theory: a holomorphic singular foliation by curves of the complex projective plane CP(2)\mathbb{CP}(2) of degree mm (given in affine coordinates by the one-form −(q+yr)dx+(p+xr)dy-(q+yr)dx+(p+xr)dy with p,q,rp,q,r polynomials, max⁡(deg⁡p,deg⁡q,deg⁡r)=m\max(\deg p,\deg q,\deg r)=m and rr homogeneous or zero) can carry invariant algebraic curves of degree at most m+2m+2: Cerveau and Lins Neto (1991) proved this for nodal curves, where all singularities are normal crossings of exactly two transversal branches, with the extra rigidity that degree m+2m+2 forces the curve to be reducible and the foliation to be of logarithmic type (defined by a closed rational form ∑iλi dFi/Fi\sum_i\lambda_i\, dF_i/F_i); Carnicer (1994) proved the same bound n≤m+2n\le m+2 in the non-dicritical case. A singularity of the foliation is dicritical if infinitely many leaves pass through it, and non-dicritical otherwise; an invariant algebraic curve is non-dicritical if the foliation has no dicritical singularities on it (such a curve may still be singular). Zhang (2011) combined these degree bounds with Harnack's theorem and the Giacomini–Llibre–Viano characterization by inverse integrating factors (a function VV satisfying pVx+qVy=V(px+qy)pV_x+qV_y=V(p_x+q_y); every limit cycle of the field lies in {V=0}\{V=0\}) to prove the conjecture in the nodal case (Theorem 1.1: the parity-refined Harnack bound is attained), and to solve the simple version for non-dicritical invariant curves with the weaker bound 1+m(m−1)/21+m(m-1)/2 for even mm and m(m−1)/2m(m-1)/2 for odd mm when r≡0r\equiv0, and 1+(m+1)m/21+(m+1)m/2 resp. (m+1)m/2(m+1)m/2 when r≢0r\not\equiv0 (Theorem 1.2); under the extra assumption that the vector field has no dicritical singularities at all and has at least three invariant algebraic curves, he recovered the sharp nodal bound (Theorem 1.4). The circular case was later settled exactly by Llibre, Ramírez, Ramírez and Sadovskaia (2016): a degree-SS vector field has at most S−1S-1 invariant circles that are algebraic limit cycles, and this is sharp. The 2021 survey by Llibre and Zhang records the general conjecture as open, along with its even more basic instances: whether every quadratic system has at most one algebraic limit cycle, and whether every cubic system has at most two.

Problem Statement

Determine the exact maximal number H(m)H(m) of algebraic limit cycles of real planar polynomial vector fields of degree mm — the 'simple version' of Hilbert's 16th problem for algebraic limit cycles. The conjectured answer, posed by Llibre–Ramírez–Sadovskaia (2010) and left open by Zhang (2011), is H(m)=1+(m−1)(m−2)/2H(m)=1+(m-1)(m-2)/2; in the classes where the problem is settled (generic curves, nodal curves, and fields having no dicritical singularities at all with at least three invariant algebraic curves) the sharp value is the parity-refined Harnack bound 1+(m−1)(m−2)/21+(m-1)(m-2)/2 for even mm and (m−1)(m−2)/2(m-1)(m-2)/2 for odd mm, while in the circular class the sharp bound for degree SS is S−1S-1 (Llibre–Ramírez–Ramírez–Sadovskaia 2016). For every mm the conjecture asserts matching upper and lower bounds, i.e. that 1+(m−1)(m−2)/21+(m-1)(m-2)/2 is an upper bound for every degree-mm system and is attained by one, and it is refuted by a degree-mm system with more algebraic limit cycles than conjectured; it is currently open even for m=2m=2 (at most one algebraic limit cycle per quadratic system) and m=3m=3 (at most two per cubic system). Zhang also explicitly left open two decisive sub-questions in the non-dicritical class: (1) decide whether there exists a holomorphic singular foliation of degree mm in CP(2)\mathbb{CP}(2) having a non-dicritical irreducible invariant algebraic curve of degree m+1m+1 or m+2m+2 (Zhang conjectures nonexistence); and (2) prove or refute Zhang's conjecture that the best upper bound on the number of algebraic limit cycles for vector fields whose invariant algebraic curves are all non-dicritical coincides with the nodal bound of his Theorem 1.1, i.e., that the bounds of his Theorem 1.2 (1+m(m−1)/21+m(m-1)/2 / m(m−1)/2m(m-1)/2 for r≡0r\equiv0 and 1+(m+1)m/21+(m+1)m/2 / (m+1)m/2(m+1)m/2 for r≢0r\not\equiv0) are not sharp.

References

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