Polynomial systems: a lower bound for the Hilbert numbers

Abstract

Abstract Let Hn be the maximum possible number of limit cycles of systems ẋ = P(x, y), ẏ = Q(x, y), where P and Q are polynomials of degree at most n. We are concerned with the rate of growth of Hn as n increases: it is known that Hn ≽ kn2 for some constant k. In this paper we show that Hn grows at least as rapidly as n2 log n.