Weak infinitesimal Hilbert’s 16th problem |
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Authors: | I A Khovanskaya |
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Institution: | (1) State University-Higher School of Economics, ul. Myasnitskaya 20, Moscow, 101000, Russia |
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Abstract: | The following weak infinitestimal Hilbert’s 16th problem is solved. Given a real polynomial H in two variables, denote by M(H, m) the maximal number possessing the following property: for any generic set {γ i } of at most M(H,m) compact connected components of the level lines H = c i of the polynomial H, there exists a form θ = P dx + Q dy with polynomials P and Q of degrees no greater than m such that the integral ∫ H=c θ has nonmultiple zeros on the connected components {γ i }. An upper bound for the number M(H,m) in terms of the degree n of the polynomial H is found; this estimate is sharp for almost every polynomial H of degree n. A multidimensional version of this result is proved. The relation between the weak infinitesimal Hilbert’s 16th problem and the following question is discussed: How many limit cycles can a polynomial vector field of degree n have if it is close to a Hamiltonian vector field? |
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