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  "body": "\n## Problem\n\nThe problem has two parts.\nThe real points of a nonsingular real plane curve form finitely many closed curves in the real projective plane.\nThose that bound a disc are called ovals.\n\n1. For each $n$, determine the possible relative positions of the ovals of a nonsingular real [[plane algebraic curve|EWS4WFTA7W]] of degree $n$. Ask the same for real algebraic surfaces in space.\n2. Consider planar vector fields $\\dot{x} = P(x, y)$, $\\dot{y} = Q(x, y)$ with $P, Q$ real polynomials of degree at most $n$. For each $n$, bound the number $H(n)$ of their [[limit cycles|0AAQ73WRFB]], and describe their possible positions.\n\n## Remarks\n\nHilbert included the problem in his list of 1900.\nFor part 1, Harnack's curve theorem says that a real plane curve of degree $n$ has at most $(n-1)(n-2)/2 + 1$ connected components, and this bound is attained.\nGudkov completed the classification of the arrangements for degree $6$, and the classification is known up to degree $7$ and open beyond.\nFor part 2, Écalle and Ilyashenko proved independently that each single polynomial vector field has finitely many limit cycles.\nWhether $H(n)$ is finite is unknown for every $n \\ge 2$, and quadratic vector fields with $4$ limit cycles are known.\n\n## Explanations\n\n### L1 human\n\nA real polynomial curve in the plane can consist of several closed loops, some nested inside others.\nThe first part asks which nesting patterns can occur for a given degree.\nThe second part asks how many isolated closed orbits a polynomial system of differential equations in the plane can have.\nBoth questions are easy to state and remain largely open after more than a century.\n",
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