Infinitesimal Hilbert 16th Problem and Lienard Equation

Zhifen Zhang

School of Mathematical Sciences

Peking University

Beijing 100871, China

Abstract. For a polynomial differential vector field, how many limit cycles it may have and what is their relative positions are known as the second part of Hilbert's 16th problem, which appears to be the most persistent one in Hilbert's 23 problems, so it is reasonable to try to start from some of its restricted versions. Infinitesimal Hilbert 16th Problem and Lienard Equation are two of its restricted versions.

Infinitesimal Hilbert 16th Problem. Given H(x, y) = h, = Q(x, y) dx - P(x, y) dy, where H , Q and P are real polynomials of x and y with deg P = deg Q = deg H - 1 = n . Let r(h) be the compact component of H = h .

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Problem: # ( I = O ) < ?

# ( I = 0) is closely related to # (L. C.) of d H + = 0.

Lienard Equation.

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Problem: # (L. C.) < ?

Both of the above problems are transformed to estimating the number of zeros of analytic functions by the generalized Jenson formula.