A perturbed differential resultant based implicitization algorithm for linear DPPEs

Let \bbK be an ordinary differential field with derivation $\partial$. Let \cP be a system of $n$ linear differential polynomial parametric equations in $n-1$ differential parameters with implicit ideal \id. Given a nonzero linear differential polynomial $A$ in \id we give necessary and sufficient conditions on $A$ for \cP to be $n-1$ 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

Determining Critical Points of Handwritten Mathematical Symbols Represented as Parametric Curves

We consider the problem of computing critical points of plane curves represented in a finite orthogonal polynomial basis. This is motivated by an approach to the recognition of hand-written mathematical symbols in which the initial data is in such an orthogonal basis and it is desired to avoid ill-conditioned basis conversions. Our main contribution is to assemble the relevant mathematical tools to perform all the necessary operations in the orthogonal polynomial basis. These include implicitization, differentiation, root finding and resultant computation

Syzygies and implicitization of tensor product surfaces

A tensor product surface is the closure of the image of a rational map λ : P1 ×P1-->P3. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of λ in P3. Currently, syzygies and Rees algebras provide the fastest and most versatile method to find implicit equations of parameterized surfaces. Knowing the structure of the syzygies of the polynomials that define the map λ allows us to formulate faster algorithms for implicitization of these surfaces and also to understand their singularities. We show that for tensor product surfaces without basepoints, the existence of a linear syzygy imposes strong conditions on the structure of the syzygies that determine the implicit equation. For tensor product surfaces with basepoints we show that the syzygies that determine the implicit equation of λ are closely related to the geometry of the set of points at which λ is undefined