Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCubic post-critically finite polynomials defined over $mathbb{Q}$
173
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We describe and implement an algorithm to find all post-critically finite (PCF) cubic polynomials defined over $mathbb{Q}$, up to conjugacy over $text{PGL}_2(bar{mathbb{Q}})$. We describe normal forms that classify equivalence classes of cubic polynomials while respecting the field of definition. Applying known bounds on the coefficients of post-critically bounded polynomials to these normal forms simultaneously at all places of $mathbb{Q}$, we create a finite search space of cubic polynomials over $mathbb{Q}$ that may be PCF. Using a computer search of these possibly PCF cubic polynomials, we find fifteen which are in fact PCF.
rate research
Read More
A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatous classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.
Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{frac{q^2 -1}{3}+1} +x$ over $mathbb{F}_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form $(x^{q} +bx + c)^{frac{q^2 -1}{d}+1} -bx$ over $mathbb{F}_{q^2}$, where $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--230]. In this paper we concentrate our efforts on the PPs of more general form [ f(x)=(ax^{q} +bx +c)^r phi((ax^{q} +bx +c)^{(q^2 -1)/d}) +ux^{q} +vx~~text{over $mathbb{F}_{q^2}$}, ] where $a,b,c,u,v in mathbb{F}_{q^2}$, $r in mathbb{Z}^{+}$, $phi(x)in mathbb{F}_{q^2}[x]$ and $d$ is an arbitrary positive divisor of $q^2-1$. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary--Ghioca--Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether $f(x)$ permutes $mathbb{F}_{q^2}$ to that of verifying whether two more polynomials permute two subsets of $mathbb{F}_{q^2}$. As a consequence, we find a series of simple conditions for $f(x)$ to be a PP of $mathbb{F}_{q^2}$. These results unify and generalize some known classes of PPs.
Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height hat{h}_f is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of f. Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on d. A related conjecture claims that at non-preperiodic rational points, hat{h}_f is bounded below by a positive constant (depending only on d) times some kind of height of f itself. In this paper, we provide support for these conjectures in the case d=3 by computing the set of small height points for several billion cubic polynomials.
Let $f:{mathbb P}^nto{mathbb P}^n$ be a morphism of degree $dge2$. The map $f$ is said to be post-critically finite (PCF) if there exist integers $kge1$ and $ellge0$ such that the critical locus $operatorname{Crit}_f$ satisfies $f^{k+ell}(operatorname{Crit}_f)subseteq{f^ell(operatorname{Crit}_f)}$. The smallest such $ell$ is called the tail-length. We prove that for $dge3$ and $nge2$, the set of PCF maps $f$ with tail-length at most $2$ is not Zariski dense in the the parameter space of all such maps. In particular, maps with periodic critical loci, i.e., with $ell=0$, are not Zariski dense.
In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterizations of permutation polynomials.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
