Simple exponential estimate for the number of real zeros of complete Abelian integrals

We show that for a generic polynomial H = H(x,y) and an arbitrary differential 1-form w = P(x, y) dx+Q(x, y) dy with polynomial coefficients of degree less than or equal to d, the number of ovals of the foliation H = const, which yield the zero value of the complete Abelian integral I(t) = closed integral(H=t) w, grows at most as exp O-H(d) as d --> infinity, where O-H(d) depends only on H. The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let f(1)(t),..., f(n)(t), t is an element of K double left hook R, be a fundamental system of real solutions to a linear ordinary differential equation Lu = 0 with rational coefficients and without singularities on the interval K. If the differential operator L is irreducible, then any real function representable in the form [GRAPHICS] with polynomial coefficients p(jk) is an element of C[t] of degree less or equal to d, may have a most exp O-L,O-K (d) real isolated zeros on K as d --> infinity.