# Polynomial Systems: A Lower Bound for the Hilbert Numbers

C. J. Christopher, N. G. Lloyd

## Abstract

Let H$_{n}$ be the maximum possible number of limit cycles of systems $\dot{x}$ = P(x, y), $\dot{y}$ = Q(x, y), where P and Q are polynomials of degree at most n. We are concerned with the rate of growth of H$_{n}$ as n increases: it is known that H$_{n}\geq$ kn$^{2}$ for some constant k. In this paper we show that H$_{n}$ grows at least as rapidly as n$^{2}$ log n.