Orthogonal polynomials on a class of planar algebraic curves

Systems of Orthogonal Polynomials on Certain Algebraic Curves

Symmetries and similarities of planar algebraic curves using harmonic polynomials

We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a similarity transformation. Both algorithms are based on the fact, well-known in Harmonic Analysis, that the Laplacian operator commutes with orthogonal transformatio...

A New Class of Orthogonal Polynomials on the Unit Circle

Even though the theory of orthogonal polynomials on the unit circle, also known as the theory of Szegő polynomials, is very extensive, it is less known than the theory of orthogonal polynomials on the real line. One reason for this may be that “beautiful” examples on the theory of Szegő polynomials are scarce. This is in contrast to the wonderful examples of Jacobi, Laguerrer and Hermite polyno...

On the complexity of computing with planar algebraic curves

In this paper, we give improved bounds for the computational complexity of computing with planar algebraic curves. More specifically, for arbitrary coprime polynomials f , g ∈ Z[x, y] and an arbitrary polynomial h ∈ Z[x, y], each of total degree less than n and with integer coefficients of absolute value less than 2 , we show that each of the following problems can be solved in a deterministic ...

Primitive Geometric Operations on Planar Algebraic Curves with Gaussian Approximation

We present a curve approximation method which approximates each planar algebraic curve segment by discrete curve points at each of which the curve has its gradient from a set of uniformly distributed normals. This method, called Gaussian Approximation (GAP), provides efficient algorithms for various primitive geometric operations, especially for those related with gradients such as common tange...