Related papers: Degree Complexity of a Family of Birational Maps
We compute the degree complexity of the family of birational maps considered in \cite{bedford-kim-tuyen-abarenkova-maillard} for all exceptional cases. Some interesting properties of the family are also given.
Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines,…
Let $f$ be a birational map of ${\bf C}^d$, and consider the degree complexity, or asymptotic degree growth rate $\delta(f)=\lim_{n\to\infty}({\rm deg}(f^n))^{1/n}$. We introduce a family of elementary maps, which have the form $f=L\circ…
We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings, based on Halburd's method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer…
This article studies the sequence of iterative degrees of a birational map of the plane. This sequence is known either to be bounded or to have a linear, quadratic or exponential growth. The classification elements of infinite order with a…
For $q\geq 3$, we let $\mathcal{S}_q$ denote the projectivization of the set of symmetric $q\times q$ matrices with coefficients in $\mathbb{C}$. We let $I(x)=(x_{i,j})^{-1}$ denote the matrix inversion, and we let $J(x)=(x_{i,j}^{-1})$ be…
We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are…
We first study birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of $ 3 \times 3 $ matrices. We introduce a semi-numerical analysis which enables to compute the Arnold complexities…
We construct a family of birational maps acting on two dimensional projective varieties, for which the growth of the degrees of the iterates is cubic. It is known that this growth can be bounded, linear, quadratic or exponential for such…
We study bipartite maps on the plane with one infinite face and one face of perimeter 2. At first we consider the problem of their enumeration an then study the connection between the combinatorial structure of a map and the degree of its…
In this paper, we define an ordering relation for a set of complex numbers, and research the properties and theorems of the ordering, solve some simple complex inequalities with the ordering.
We introduce the topological complexity of the work map associated to a robot system. In broad terms, this measures the complexity of any algorithm controlling, not just the motion of the configuration space of the given system, but the…
We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…
We consider the set of all 2-step recurrences (difference equations) that are given by linear fractional maps. These give birational maps of the plane. We determine the degree growth of these birational maps. We find the all the maps in…
This paper classifies the complexity of various teaching models by their position in the arithmetical hierarchy. In particular, we determine the arithmetical complexity of the index sets of the following classes: (1) the class of uniformly…
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.
We compute the higher topological complexity of ordered configuration spaces of orientable surfaces, thus extending Cohen-Farber's description of the ordinary topological complexity of those spaces.
We show that the ordinal of the dynamical degrees of all complex birational maps of the projective plane is $\omega^\omega$.
We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…
This paper presents a combinatorial analog of topological complexity for finite spaces. We demonstrate that this coincides with the genuine topological complexity of the original finite space, and constitutes an upper bound for the…