On the regularity of a semialgebraic function
Volume 125 / 2020
Annales Polonici Mathematici 125 (2020), 25-46
MSC: Primary 14P10; Secondary 14P20, 32B20, 58A07.
DOI: 10.4064/ap190719-19-3
Published online: 15 June 2020
Abstract
Let be a semialgebraic function of class C^1, defined on an open set {U \subset \mathbb R ^n}. Let P(x,y) \in \mathbb R [x_1,\dots , x_n, y] be a polynomial of degree d such that {P(x,f(x)) = 0}, x\in U. We prove that if f is of class C^K with K \gt \frac {1}{2}d^7, then f is analytic. If n=1, then it suffices that K \gt \frac {1}{2}d^2.
