80 research outputs found
A perturbed differential resultant based implicitization algorithm for linear DPPEs
Let \bbK be an ordinary differential field with derivation . Let
\cP be a system of linear differential polynomial parametric equations in
differential parameters with implicit ideal \id. Given a nonzero linear
differential polynomial in \id we give necessary and sufficient
conditions on for \cP to be dimensional. We prove the existence of
a linear perturbation \cP_{\phi} of \cP so that the linear complete
differential resultant \dcres_{\phi} associated to \cP_{\phi} is nonzero. A
nonzero linear differential polynomial in \id is obtained from the lowest
degree term of \dcres_{\phi} and used to provide an implicitization algorithm
for \cP
Rational Hausdorff Divisors: a New approach to the Approximate Parametrization of Curves
In this paper we introduce the notion of rational Hausdorff divisor, we
analyze the dimension and irreducibility of its associated linear system of
curves, and we prove that all irreducible real curves belonging to the linear
system are rational and are at finite Hausdorff distance among them. As a
consequence, we provide a projective linear subspace where all (irreducible)
elements are solutions to the approximate parametrization problem for a given
algebraic plane curve. Furthermore, we identify the linear system with a plane
curve that is shown to be rational and we develop algorithms to parametrize it
analyzing its fields of parametrization. Therefore, we present a generic answer
to the approximate parametrization problem. In addition, we introduce the
notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve
can always be parametrized with a generic rational parametrization having
coefficients depending on as many parameters as the degree of the input curve
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