It is easy to know that the conic h(x, y) = [x.sup.2] + [cy.sup.2] - 1, c [member of] R, is an invariant

algebraic curve of systems

As an intermediate step towards the third result, we also show that for a fixed family of plane

algebraic curves with s degrees of freedom, every set of n points in the plane has a subset of [OMEGA]([n.sup.1-1/s]) points incident to a single curve, or a subset of [OMEGA]([n.sup.1/s]) points such that at most s of them lie on a curve.

Hilbert's 16th problem has two parts and its first part is on the topology of

algebraic curves and surfaces.

A projective plane

algebraic curve over C is defined as the set

Via the correspondence between

algebraic curves and floor diagrams [BM09, Theorem 2.5] these notions correspond literally to the respective analogues for

algebraic curves.

To compute focus quantities of (1.5) we first look for the branch of the

algebraic curve [[PSI].sup.(n)] (z,w) passing through the origin (that is, for a function w = f (z) of the form (1.1) such that [[PSI].sup.(n)] (z,f (z)) [equivalent to] 0).

The investigation of an

algebraic curve of the type exemplified in equation (1) has several potential uses:

For any irreducible plane

algebraic curve A [subset] [P.sub.2] the punctured Riemann surface [Mathematical Expression Omitted] is hyperbolic, in particular any holomorphic map [Mathematical Expression Omitted] (which may also be reducible) is constant.

Consider the complex

algebraic curve (the real surface) [summation] in [C.sup.2] whose equation is {[w.sup.2] = z}, a complex parabola.

Mathematicians from Europe and the US discuss geometry, algebraic geometry, and topology, including the basic properties of uniformly rational varieties, new relations between algebraic topology and the theory of Hopf, a measured foliated 2-complex thin type, the classical geometry of complexes, a generalization of the amoeba and the Ronkin function of a plane

algebraic curve for a pair of harmonic functions, and natural differential-geometric constructions on the algebra of densities.

These papers provide a powerful method to find a presentation of the fundamental group of (the complement of) any

algebraic curve C in C[P.sup.2].