On certain maximal hyperelliptic curves related to Chebyshev polynomials

We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs such that the hyperelliptic curve over a finite field given by is maximal over the finite field of cardinality . Here denotes the Chebyshev polynomial of degree d. The same question is studied for the curves given by , and also for . Our results generalize some of the statements in [12]203276293FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP2017/19190-5The first author was supported by FAPESP/SP-Brazil grant 2017/19190-5. The second author thanks Marco Streng and Nurdagül Anbar Meidl for helpful suggestions. We also thank the referee for various comments which helped us improve the expositio

Trivariate polynomial approximation on Lissajous curves

We study Lissajous curves in the 3-cube, that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct trivariate hyperinterpolation polynomials via a single 1-d Fast Chebyshev Transform (by the Chebfun package), and to compute discrete extremal sets of Fekete and Leja type for trivariate polynomial interpolation. Applications could arise in the framework of Lissajous sampling for MPI (Magnetic Particle Imaging)

On the syzygies and Alexander polynomials of nodal hypersurfaces

We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining a nodal hypersurface. The result gives information on the position of the singularities of a nodal hypersurface expressed in terms of defects or superabundances. The case of Chebyshev hypersurfaces is considered as a test for this result and leads to a potentially infinite family of nodal hypersurfaces having nontrivial Alexander polynomials.Comment: The second version: some minor changes made and Example 4.3 involving Kummer surfaces with 16 nodes adde

Prime and composite Laurent polynomials

In 1922 Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly indecomposable polynomial solutions of the functional equation f(p(z))=g(q(z)). In this paper we study the equation above in the case when f,g,p,q are holomorphic functions on compact Riemann surfaces. We also construct a self-contained theory of functional decompositions of rational functions with at most two poles generalizing the Ritt theory. In particular, we give new proofs of the theorems of Ritt and of the theorem of Bilu and Tichy.Comment: Some of the proofs given in sections 6-8 are simplified. Some other small alterations were mad

Green's functions for multiply connected domains via conformal mapping

A method is described for the computation of the Green's function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz-Christoffel conformal map of the part of the upper half-plane exterior to the problem domain onto a semi-infinite strip whose end contains K-1 slits. From the Green's function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iteration. By making the end of the strip jagged, the method can be generalised to weighted Green's functions and weighted approximations

Polynomial Meshes: Computation and Approximation

We present the software package WAM, written in Matlab, that generates Weakly Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d polynomial least squares and interpolation on compact sets with various geometries. Possible applications range from data fitting to high-order methods for PDEs