{"id":"openai-math/uniform-bounds-for-planar-polynomial-limit-cycles","pdf":"https://github.com/openai/math/blob/main/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026","tags":["dynamical-systems","limit-cycles","hilbert-sixteenth-problem","polynomial-vector-fields"],"score":0,"title":"Uniform bounds for planar polynomial limit cycles","domain":"Dynamical systems and ergodic theory","family":"143","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DS","summary":"For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-24","reactions":{},"discussion":[{"url":"https://www.youtube.com/watch?v=QNSbI3mrrhc","kind":"web","site":"youtube.com","text":"Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube.","added":"2026-10-24","title":"OpenAI Claimed 722 Math Breakthroughs. Did Any Hold Up?","author":"Nontrivial"}],"formalized":false,"family_title":"Hilbert's sixteenth problem: uniform bounds for limit cycles"}