We provide an algorithm which decides whether a polynomial foliation F C 2 on the complex plane has a polynomial first integral of genus g = 1.Except in a specific case, an extension of the algorithm also decides if F C 2 has a rational first integral of that genus.

We provide a lower bound on the degree of curves of the projective plane P 2 passing through the centers of a divisorial valuation ν of P 2 with prescribed multiplicities, and an upper bound for the Seshadri-type constant of ν, μ(ν), constant that is crucial in the Nagata-type valuative conjecture. We also give some results related to the bounded negativity conjecture concerning those rational surfaces having the projective plane as a relatively minimal model.

We provide a lower bound on the degree of curves of the projective plane $\mathbb {P}^2$ passing through the centers of a divisorial valuation $\nu $ of $\mathbb {P}^2$ with prescribed multiplicities, and an upper bound for the Seshadri-type constant of $\nu $, $\hat {\mu }(\nu )$, constant that is crucial in the Nagata-type valuative conjecture. We also give some results related to the bounded negativity conjecture concerning those rational surfaces having the projective plane as a relatively minimal model.

We study algebraic integrability of complex planar polynomial vector fields X = A(x, y)(∂/∂x) + B(x, y)(∂/∂y) through extensions to Hirzebruch surfaces. Using these extensions, each vector field X determines two infinite families of planar vector fields that depend on a natural parameter which, when X has a rational first integral, satisfy strong properties about the dicriticity of the points at the line x = 0 and of the origin. As a consequence, we obtain new necessary conditions for algebraic integrability of planar vector fields and, if X has a rational first integral, we provide a region in R 2 ≥0 that contains all the pairs (i, j) corresponding to monomials x i y j involved in the generic invariant curve of X.