Polynomial shift-like maps in ℂk

The purpose of this paper is to explore a few properties of polynomial shift-like automorphisms of [Formula: see text] We first prove that a [Formula: see text]-shift-like polynomial map (say [Formula: see text]) degenerates essentially to a polynomial map in [Formula: see text]-dimensions as [Formula: see text] Second, we show that a shift-like map obtained by perturbing a hyperbolic polynomial (i.e. [Formula: see text], where [Formula: see text] is sufficiently small) has finitely many Fatou components, consisting of basins of attraction of periodic points and the component at infinity.

On the connectivity of the hyperbolicity region of irreducible polynomials

We give a proof for the fact that an irreducible hyperbolic polynomial has only one pair of hyperbolicity cones. Apart from the use of Bertini’s Theorem the proof is elementary.

Symbolic computation in hyperbolic programming

Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming (SDP). We propose an approach based on symbolic computation, relying on the multiplicity structure of the algebraic boundary of the cone, without the assumption of determinantal representability. This allows us to design exact algorithms able to certify the multiplicity of the solution and the optimal value of the linear function.

A continuity result on quadratic matings with respect to parameters of odd denominator rationals

In this paper we prove a continuity result on matings of quadratic lamination maps sp depending on odd denominator rationals p ∈(0,1). One of the two mating components is fixed in the result. Note that our result has its implication on continuity of matings of quadratic hyperbolic polynomials fc(z)=z2 + c, c ∈ M the Mandelbrot set with respect to the usual parameters c. This is because every quadratic hyperbolic polynomial in M is contained in a bounded hyperbolic component. Its center is Thurston equivalent to some quadratic lamination map sp, and there are bounds on sizes of limbs of M and on sizes of limbs of the mating components on the quadratic parameter slice Perm′(0).