How many ovals can a plane curve have, and how can they nest? A zoo of wild curves ending at an open problem
Draw the real solutions of a polynomial and you get a curve, often with closed loops called ovals. How many ovals is settled. How they can nest is Hilbert’s sixteenth problem, and it is still open.
Browse eight classical curves. On the Cassini ovals, slide the parameter and watch two ovals fuse into one, passing through the lemniscate.
The figure-eight Jacob Bernoulli published in 1694, describing it as a ribbon; the points where the product of the distances to two fixed foci equals the square of half the distance between them.
Every one of these is the zero set of a polynomial in x and y — no trigonometry, no parametrisation, just an equation. The petals, cusps, loops, and ovals are all consequences of algebra. Cusps are where the curve is singular, which is why the astroid and deltoid need higher degree than their smooth-looking outlines suggest.
Multiply a few ellipses together, then add a small constant ε to burst every crossing open. Sweep ε through zero and watch the arrangement and oval count flip.
Every node has resolved into a small arc, and the sign of ε decides which of the two ways each one opens. That single choice is the difference between 9 ovals and 5 — same curve, same degree, wildly different topology.
Harnack’s theorem (1876): a smooth real curve of degree d has at most (d−1)(d−2)/2 + 1 connected components, and the bound is always attained. Curves that reach it are called M-curves. The bound is the easy half of the story — the hard half, which is still open, is how those ovals are allowed to be arranged.
A smooth real projective curve breaks into finitely many connected pieces. In the projective plane, a piece either bounds a disc — an oval — or it does not, and there can be at most one of the latter kind. So counting pieces is essentially counting ovals, and it is a genuine topological invariant of the curve.
Harnack’s theorem (1876): a smooth curve of degree d has at most (d−1)(d−2)/2 + 1 components. The expression is not arbitrary — it is the genus of a smooth plane curve of degree d, plus one. The complex curve is a surface with g handles, complex conjugation acts on it, and the real curve is the fixed set; a theorem about how an involution can act on a genus-g surface caps the number of fixed circles at g + 1.
Harnack also proved the bound is sharp at every degree, by constructing curves that reach it. Those are the M-curves — maximal curves. His method is the perturbation trick in the next demo: build a deliberately singular curve as a product of simpler ones, then nudge it so every singular point bursts open.
Compare the three ways to arrange eleven ovals on a sextic, found over seventy-eight years, then see how far the classification has got.
Harnack proved the bound of 11 ovals for a sextic and built a curve attaining it, with nine ovals lying free and one more tucked inside an eleventh.
All three of these are smooth curves of degree 6 with the maximum 11 ovals. Harnack settled how many ovals there can be in 1876. But how they may be nested is a different question entirely, and it is the one Hilbert put on his famous list in 1900 as the sixteenth problem. He believed only two arrangements were possible. Gudkov found the third in 1969 and proved that the list stops at three.
The main tool for building curves with prescribed ovals is Viro’s patchworking (1979): glue curves together from combinatorial pieces, in a construction closely related to the tropical geometry in this module’s amoebas lesson. Patchworking also demolished the Ragsdale conjecture of 1906 — Itenberg used it in the 1990s to build counterexamples, more than eighty years after the conjecture was posed.
Knowing a sextic can have 11 ovals says nothing about how they sit relative to one another. Ovals can nest — one enclosing another — and the pattern of nesting is what Hilbert asked about. For degree 6 with the maximal 11 ovals there turn out to be exactly three possibilities, and getting to that answer took nearly a century.
Harnack exhibited one arrangement in 1876. Hilbert found a second in 1891 and believed that was the end of it. He was wrong: Gudkov constructed a third in 1969 and proved the list complete. Gudkov also spotted a startling pattern in the data — for an M-curve of even degree d = 2k, the number of ovals of one nesting parity satisfies a congruence mod 8 — and that observation, later proved by Arnold and Rokhlin, opened up the whole modern approach via topology of four-manifolds.
The other great surprise was a demolition. In 1906 Ragsdale conjectured bounds on how ovals of each parity could be distributed. It stood for most of a century before Itenberg produced counterexamples in the 1990s, built with Viro’s patchworking — a technique for assembling curves out of combinatorial pieces that is a close cousin of the tropical geometry in this module’s amoebas lesson. The stick figures turned out to be strong enough to break a ninety-year-old conjecture.
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