  Given two polynomials $P,q$ we consider the following question: "how large
can the index of the first non-zero moment $\tilde{m}_k=\int_a^b P^k q$ be,
assuming the sequence is not identically zero?". The answer $K$ to this
question is known as the moment Bautin index, and we provide the first general
upper bound: $K\leqslant 2+\mathrm{deg} q+3(\mathrm{deg} P-1)^2$. The proof is
based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals
of certain algebraic functions.
  The moment Bautin index plays an important role in the study of bifurcations
of periodic solution in the polynomial Abel equation $y'=py^2+\varepsilon qy^3$
for $p,q$ polynomials and $\varepsilon \ll 1$. In particular, our result
implies that for $p$ satisfying a well-known generic condition, the number of
periodic solutions near the zero solution does not exceed $5+\mathrm{deg}
q+3\mathrm{deg}^2 p$. This is the first such bound depending solely on the
degrees of the Abel equation.
