Math 2.0

Algebraic geometry Real algebraic geometry Problem

Hilbert's sixteenth problem

Builds on Plane curveLimit cycleReal numbers

Problem

The problem has two parts. The real points of a nonsingular real plane curve form finitely many closed curves in the real projective plane. Those that bound a disc are called ovals.

  1. For each nn, determine the possible relative positions of the ovals of a nonsingular real plane algebraic curve of degree nn. Ask the same for real algebraic surfaces in space.
  2. Consider planar vector fields x˙=P(x,y)\dot{x} = P(x, y), y˙=Q(x,y)\dot{y} = Q(x, y) with P,QP, Q real polynomials of degree at most nn. For each nn, bound the number H(n)H(n) of their limit cycles, and describe their possible positions.

Remarks

Hilbert included the problem in his list of 1900. For part 1, Harnack's curve theorem says that a real plane curve of degree nn has at most (n−1)(n−2)/2+1(n-1)(n-2)/2 + 1 connected components, and this bound is attained. Gudkov completed the classification of the arrangements for degree 66, and the classification is known up to degree 77 and open beyond. For part 2, Écalle and Ilyashenko proved independently that each single polynomial vector field has finitely many limit cycles. Whether H(n)H(n) is finite is unknown for every n≥2n \ge 2, and quadratic vector fields with 44 limit cycles are known.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Limit cycle, concept, no evidence levelLimit cycleconceptPlane curve, concept, no evidence levelPlane curveconceptReal numbers, object, no evidence levelReal numbersobjectHilbert's sixteenth problem, problem, no evidence levelHilbert's sixteenth p…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit cycle concept
  2. Plane curve concept
  3. Real numbers object
  4. Hilbert's sixteenth problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_sixteenth_problem
About this page
TagC1XNMMKZ1J
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Kindproblem
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MSC14P25, 34C07
StatusThis copy of the site has no build data. Run m2 status C1XNMMKZ1J.
AboutPlane curve, Limit cycle
UsesPlane curve, Limit cycle, Real numbers
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