Real plane algebraic curves with prescribed singularities

Real Plane Algebraic Curves

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi–definite nondefinite or definite. We present a discussion about isolated points. By means of the operator, these points can ...

Plane Curves of Minimal Degree with Prescribed Singularities

We prove that there exists a positive α such that for any integer d ≥ 3 and any topological types S1, . . . , Sn of plane curve singularities, satisfying μ(S1) + · · ·+ μ(Sn) ≤ αd, there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S1, . . . , Sn, respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove tha...

An asymptotic existence theorem for plane curves with prescribed singularities

Let d,m1, . . . ,mr be (r + 1) positive integers. Denote by V (d;m1, . . . ,mr) the variety of irreducible (complex) plane curves of degree d having exactly r ordinary singularities of multiplicities m1, . . . ,mr. In most cases, it is still an open problem to know whether this variety is empty or not. In this paper, we will concentrate on the case where the r singularities can be taken in a ge...

On the Existence of Plane Curves with Prescribed Singularities

Analysis of Real Algebraic Plane Curves

This work describes a new method to compute geometric properties of a real algebraic plane curve of arbitrary degree. These properties contain the topology of the curve as well as the location of singular points and vertical asymptotes. The algorithm is based on the Bitstream Descartes method (Eigenwillig et al.: “A Descartes Algorithm for Polynomials with Bit-Stream Coefficients”, LNCS 3718), ...