Ouchi, Koji (2006-12). Exact polynomial system solving for robust geometric computation. Doctoral Dissertation. Thesis uri icon

abstract

  • I describe an exact method for computing roots of a system of multivariate polynomials with rational coefficients, called the rational univariate reduction. This method enables performance of exact algebraic computation of coordinates of the roots of polynomials. In computational geometry, curves, surfaces and points are described as polynomials and their intersections. Thus, exact computation of the roots of polynomials allows the development and implementation of robust geometric algorithms. I describe applications in robust geometric modeling. In particular, I show a new method, called numerical perturbation scheme, that can be used successfully to detect and handle degenerate configurations appearing in boundary evaluation problems. I develop a derandomized version of the algorithm for computing the rational univariate reduction for a square system of multivariate polynomials and a new algorithm for a non-square system. I show how to perform exact computation over algebraic points obtained by the rational univariate reduction. I give a formal description of numerical perturbation scheme and its implementation.
  • I describe an exact method for computing roots of a system of multivariate
    polynomials with rational coefficients, called the rational univariate reduction. This
    method enables performance of exact algebraic computation of coordinates of the
    roots of polynomials. In computational geometry, curves, surfaces and points are described
    as polynomials and their intersections. Thus, exact computation of the roots
    of polynomials allows the development and implementation of robust geometric algorithms.
    I describe applications in robust geometric modeling. In particular, I show
    a new method, called numerical perturbation scheme, that can be used successfully
    to detect and handle degenerate configurations appearing in boundary evaluation
    problems. I develop a derandomized version of the algorithm for computing the rational
    univariate reduction for a square system of multivariate polynomials and a
    new algorithm for a non-square system. I show how to perform exact computation
    over algebraic points obtained by the rational univariate reduction. I give a formal
    description of numerical perturbation scheme and its implementation.

publication date

  • December 2006