For a nonsingular real algebraic curve of even degree 2k in the real projective plane, let p and n denote the numbers of even and odd ovals. Ragsdale's 1906 bounds on p and n, in their refined form p, n ≤ (3k²−3k)/2 + 1, were disproved for arbitrary curves by the patchworking constructions of Itenberg and Haas, and Brugallé (2006) then built curves with asymptotically (7/4)k² even (resp. odd) ovals, matching the Petrovsky–Harnack upper bounds. However, no counterexample is known among M-curves — curves attaining the maximal number 2k²−3k+2 of connected components — and the only partial result in the maximal class is Haas' theorem for M-curves produced by unimodular combinatorial patchworking (p ≤ (3k²−3k)/2 + 1, n ≤ (3k²−3k)/2 + 4). The question is open as of 2026.

Background

The real part of a nonsingular real algebraic curve of even degree 2k2k in the real projective plane RP2\mathbb{RP}^2 is a disjoint union of circles, each of which separates RP2\mathbb{RP}^2 and is therefore called an oval; the disk it bounds is its interior. An oval is even (respectively odd) if it lies inside an even (respectively odd) number of other ovals. Writing pp and nn for the numbers of even and odd ovals, the first part of Hilbert's sixteenth problem — the classification, up to isotopy, of the possible arrangements of the real part of a curve of given degree — reduces, for even degree, to the classification of realizable arrangements of ovals, and pp and nn are the coarsest invariants of such an arrangement. By Harnack's bound, a curve of genus gg has at most g+1g+1 connected components; for degree 2k2k this gives p+n≤(2k−1)(2k−2)/2+1=2k2−3k+2p+n\le(2k-1)(2k-2)/2+1=2k^2-3k+2, and a curve attaining this bound is called an M-curve (maximal curve).

Ragsdale (1906), from an empirical study of arrangements, conjectured that p≤3k(k−1)2+1p\le\frac{3k(k-1)}{2}+1 and n≤3k(k−1)2n\le\frac{3k(k-1)}{2}, together with difference inequalities between pp and nn. Petrovsky (1933–1938) proved the difference inequalities p−n≤3k(k−1)2+1p-n\le\frac{3k(k-1)}{2}+1 and n−p≤3k(k−1)2n-p\le\frac{3k(k-1)}{2} (by an interpolation formula combined with Morse theory) and conjectured the symmetric bounds p,n≤3k(k−1)2+1p,n\le\frac{3k(k-1)}{2}+1; this symmetric statement is now called the refined Ragsdale conjecture (Bertrand–Brugallé–Renaudineau 2017). Combining Petrovsky's inequality with Harnack's bound yields the best general upper bounds

p≤7k2−9k+64andn≤7k2−9k+44.p\le\frac{7k^2-9k+6}{4}\quad\text{and}\quad n\le\frac{7k^2-9k+4}{4}.

Viro (1980) constructed M-curves of degree 8 with n=3k(k−1)2+1=19n=\frac{3k(k-1)}{2}+1=19 odd ovals, disproving Ragsdale's original one-sided bound n≤3k(k−1)2n\le\frac{3k(k-1)}{2} while attaining the refined bound exactly. For arbitrary (non-maximal) curves the refined conjecture is false: Itenberg (1993, 1995), using Viro's combinatorial patchworking (a gluing technique that assembles a real algebraic curve from polynomial pieces on a triangulation of the Newton polygon, i.e. the convex hull of the exponent vectors of the monomials), built curves with p=3k(k−1)2+1+(k−3)2+48p=\frac{3k(k-1)}{2}+1+\frac{(k-3)^2+4}{8} and similarly for nn; Haas (1995) gave the multilucarne counterexamples; Itenberg (2001) reached 81k248+O(k)\frac{81k^2}{48}+O(k) ovals of each parity; and Brugallé (2006) constructed curves with p/k2→74p/k^2\to\frac74 (and likewise n/k2→74n/k^2\to\frac74), proving that the Petrovsky–Harnack bounds above are asymptotically sharp. None of these constructions produces an M-curve, and Brugallé (2006) explicitly points out that no counterexample to the Ragsdale conjectures is known among curves with the maximal number of connected components.

For M-curves the refined conjecture takes a compact equivalent form: since p+n=2k2−3k+2p+n=2k^2-3k+2, the two bounds p,n≤3k2−3k2+1p,n\le\frac{3k^2-3k}{2}+1 hold simultaneously if and only if ∣p−n∣≤k2|p-n|\le k^2. The difference p−np-n is the quantity constrained by complex-orientation theory (the study of the action of the anti-holomorphic involution on the complexification of the curve; for an M-curve of degree 2k2k the Rokhlin congruence gives p−n≡k2(mod8)p-n\equiv k^2\pmod 8). The only known partial result towards the conjecture in the maximal class is a theorem from Haas' 1997 thesis (never published as a paper; recalled in Bertrand–Brugallé–Renaudineau 2017): if an M-curve of degree 2k2k is obtained by a unimodular combinatorial patchworking (patchworking over a unimodular triangulation, i.e. one all of whose triangles have unit normalized area), then p≤3k2−3k2+1p\le\frac{3k^2-3k}{2}+1 and n≤3k2−3k2+4n\le\frac{3k^2-3k}{2}+4; moreover, such a curve with more than 3k2−3k2+1\frac{3k^2-3k}{2}+1 odd ovals has exactly 3k2−3k2+4\frac{3k^2-3k}{2}+4 of them. Bertrand–Brugallé–Renaudineau (2017) note that this is, to their knowledge, the only known result in the direction of the Ragsdale conjecture for maximal curves, and that the conjecture remains one challenging open question in the case of M-curves. Gabard (2013) proposed an approach through total reality (Riemann-style pencils of real lines on the complexification) and the orientability of the doubled oval membrane (the Arnold surface), and reported unpublished results of Fiedler for symmetric M-curves (curves invariant under a holomorphic involution of CP2\mathbb{CP}^2): the refinement p−n≡k2(mod16)p-n\equiv k^2\pmod{16}, and the bound p−n≤k2p-n\le k^2 for symmetric M-curves with a nest of depth k−2k-2 (a chain of k−2k-2 ovals each containing the next).

Problem Statement

Decide the refined Ragsdale conjecture in the maximal class:

For every integer k≥1 and every nonsingular real algebraic M-curve A of degree 2k in RP2,  p(A)≤3k2−3k2+1 and n(A)≤3k2−3k2+1.\text{For every integer } k\ge 1 \text{ and every nonsingular real algebraic M-curve } A \text{ of degree } 2k \text{ in } \mathbb{RP}^2,\ \ p(A)\le\frac{3k^2-3k}{2}+1 \text{ and } n(A)\le\frac{3k^2-3k}{2}+1.

Here an M-curve is a curve whose real part has the maximal number 2k2−3k+22k^2-3k+2 of connected components, all of them ovals; p(A)p(A) and n(A)n(A) are the numbers of even and odd ovals. Since p+n=2k2−3k+2p+n=2k^2-3k+2 for an M-curve, the statement is equivalent to ∣p−n∣≤k2|p-n|\le k^2.

The question is settled either by a proof that the bounds hold for every M-curve of every even degree, or by an M-curve of some even degree 2k2k that violates at least one of the two bounds. The class of curves is unrestricted: any nonsingular real curve of even degree attaining the Harnack maximum, with no symmetry, genericity, or construction-method hypothesis.

References

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