Simple Exponential Estimate for the Number of Real Zeros of Complete Abelian Integrals

One of the main results of this paper is an upper bound for the total number of real isolated zeros of complete Abelian integrals, exponential in the degree of the form (Theorem 1 below). This result improves a previously obtained in [IY1] double exponential estimate for the number of real isolated zeros on a positive distance from the singular locus. In fact, the theorem on zeros of Abelian integrals is a particular case of a more general result concerning the number of zeros in polynomial envelopes of irreducible and essentially irreducible differential operators and equations (see §1.3 below). The first announcement of these and other results proved below was in [NY]. In §1 all principal results are formulated and all necessary definitions gathered, §2 explains connections between Abelian integrals and polynomial