Mădălina Hodorog defended her thesis in excellent fashion!
In computer algebra, the problem of computing topological invariants (i.e. delta-invariant, genus) of a plane complex algebraic curve is well-understood if the coefficients of the defining polynomial of the curve are exact data (i.e. integer numbers or rational numbers). The challenge is to handle this problem if the coefficients are inexact (i.e. numerical values). In this thesis, we approach the algebraic problem of computing invariants of a plane complex algebraic curve defined by a polynomial with both exact and inexact data. For the inexact data, we associate a positive real number called tolerance or noise, which measures the error level in the coefficients. We deal with an ill-posed problem in the sense that, tiny changes in the input data lead to dramatic modifications in the output solution. For handling the ill-posedness of the problem we present a regularization method, which estimates the invariants of a plane complex algebraic curve. Our regularization method consists of a set of symbolic-numeric algorithms that extract structural information on the input curve, and of a parameter choice rule, i.e. a function in the noise level. We first design the following symbolic-numeric algorithms for computing the invariants of a plane complex algebraic curve:
• we compute the link of each singularity of the curve by numerical equation solving;
• we compute the Alexander polynomial of each link by using algorithms from compu-
tational geometry (i.e. an adapted version of the Bentley-Ottmann algorithm) and
combinatorial objects from knot theory;
• we derive a formula for the delta-invariant and for the genus.
We then prove that the symbolic-numeric algorithms together with the parameter choice rule compute approximate solutions, which satisfy the convergence for noisy data property. Moreover, we perform several numerical experiments, which support the validity for the convergence statement.
We implement the designed symbolic-numeric algorithms in a new software package called Genom3ck, developed using the Axel free algebraic geometric modeler and the Mathemagix free computer algebra system. For our purpose, both of these systems provide modern graphical capabilities, and algebraic and geometric tools for manipulating algebraic curves and surfaces defined by polynomials with both exact and inexact data. Together with its main functionality to compute the genus, the package Genom3ck computes also other type of information on a plane complex algebraic curve, such as the singularities of the curve in the projective plane and the topological type of each singularity.
DK9: Symbolic-Numeric Techniques for Genus Computation and Parametrization