On the Number of Limit Cycles of a Monodromic Polynomial Vector Field on the Plane

Limit cycles of monodromic polynomial vector fields are studied. These are fields on the real plane with the unique singular point $0$ whose components are polynomials and whose phase portraits are diffeomorphic to a linear focus. The number of limit cycles of such a field is estimated from above in terms of the degrees of the polynomials, the maximum of the absolute values of their coefficients, and certain characteristics of the monodromy transformation, which can be called multipliers at zero and infinity. The estimate is based on the growth and zeros theorem for holomorphic functions.