Rational Functions Of Degree Research Articles

The minimal resultant locus

Let K be a complete, algebraically closed nonarchimedean valued field, and let f(z) in K(z) be a rational function of degree d at least 2. We give an algorithm to determine whether f(z) has potential good reduction over K, based on a geometric reformulation of the problem using the Berkovich Projective Line. We show the minimal resultant is is either achieved at a single point in the Berkovich line, or on a segment, and that minimal resultant locus is contained in the tree in spanned by the fixed points and the poles of f(z). When f(z) is defined over the rationals, the algorithm runs in probabilistic polynomial time. If f(z) has potential good reduction, and is defined over a subfield H of K, we show there is an extension L/H in K with degree at most (d + 1)^2 such that f(z) achieves good reduction over L.

Small Dynamical Heights for Quadratic Polynomials and Rational Functions

Let be a polynomial or rational function of degree 2. A special case of Morton and Silverman’s dynamical uniform boundedness conjecture states that the number of rational preperiodic points of φ is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height of a nonpreperiodic rational point x is bounded below by a uniform multiple of the height of φ itself. We provide support for these conjectures by computing the set of preperiodic and small-height rational points for a set of degree-2 maps far beyond the range of previous searches.

Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps

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 Fatou's classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.

A family of rational maps with buried Julia components

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However, all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least five. This paper introduces a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of postcritically separating Julia components is encoded by a weighted dynamical tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover, this family contains some rational maps of degree three with explicit formula that answers a question McMullen raised.

A criterion for potentially good reduction in nonarchimedean dynamics

Let K be a non-archimedean eld, and let 2 K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sucient condition, involving only the xed points of and their preimages, that determines whether or not the dynamical system : P 1 ! P 1 has potentially good reduction. Fix the following notation throughout this paper. K: a eld K: an algebraic closure of K j j v: a non-archimedean absolute value on K OK: the ring of integersfx2 K :jxjv 1g of K MK: the maximal ideafx2 K :jxjv < 1g ofOK k: the residue eld

A HASSE PRINCIPLE FOR PERIODIC POINTS

Let F be a global field, let φ ∈ F(x) be a rational map of degree at least 2, and let α ∈ F. We say that α is periodic if φn(α) = α for some n ≥ 1. A Hasse principle is the idea, or hope, that a phenomenon which happens everywhere locally should happen globally as well. The principle is well known to be true in some situations and false in others. We show that a Hasse principle holds for periodic points, and further show that it is sufficient to know that α is periodic on residue fields for every prime in a set of natural density 1 to know that α is periodic in F.

ABC implies primitive prime divisors in arithmetic dynamics

Let K be a number field, let f(x) in K(x) be a rational function of degree d> 1, and let z in K be a wandering point such that f^n(z) is nonzero for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that p | f^n(z) and p does not divide f^m(z) for all positive integers m < n. We prove the same result unconditionally for function fields of characteristic 0 when f is not isotrivial.

Rational preimages in families of dynamical systems

Let $${\phi}$$ be a rational function of degree at least two defined over a number field k. Let $${a \in \mathbb{P}^1(k)}$$ and let K be a number field containing k. We study the cardinality of the set of rational iterated preimages Preim $${(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a {\rm for some} N \geq 1\}}$$ . We prove two new results (Theorems 2 and 4) bounding $${|{\rm Preim}(\phi, a, K)|}$$ as $${\phi}$$ varies in certain families of rational functions. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for Preim $${(\phi, a, K)}$$ and prove that a version of this conjecture is implied by other well-known conjectures in arithmetic dynamics.

Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics

Let $f$ be a rational function of degree $d>1$ on the projective line over a possibly non-archimedean algebraically closed field. A well-known process initiated by Brolin considers the pullbacks of points under iterates of $f$, and produces an important equilibrium measure. We define the asymptotic Fekete property of pullbacks of points, which means that they mirror the equilibrium measure appropriately. As application, we obtain an error estimate of equidistribution of pullbacks of points for $C^1$-test functions in terms of the proximity of wandering critical orbits to the initial points, and show that the order is $O(\sqrt{kd^{-k}})$ upto a specific exceptional set of capacity 0 of initial points, which is contained in the set of superattracting periodic points and the omega-limit set of wandering critical points from the Julia set or the presingular domains of $f$. As an application in arithmetic dynamics, together with a dynamical Diophantine approximation, these estimates recover Favre and Rivera-Letelier's quantitative equidistribution in a purely local manner.

Periods of rational maps modulo primes

Let K be a number field, let $${\varphi \in K(t)}$$ be a rational map of degree at least 2, and let $${\alpha, \beta \in K}$$ . We show that if α is not in the forward orbit of β, then there is a positive proportion of primes $${\mathfrak{p}}$$ of K such that $${\alpha {\rm mod} \mathfrak{p}}$$ is not in the forward orbit of $${\beta {\rm mod} \mathfrak{p}}$$ . Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace α by a hypersurface, such as the ramification locus of a morphism $${\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}}$$ .

Normal families and composite meromorphic functions

In the paper,we study the normality of families of meromorphic functions.We prove the result: Letα(z) be a meromorphic function,F a family of meromorphic functions in a domain D,and R(z) be a rational function of degree at least 3.If R o f(z) and R o g(z) shareα(z) IM for each pair f(z),g(z)∈F and one of the following conditions holds:(1)R(z)-α(z_0) has at least three distinct zeros or poles for any z_0∈D;(2)There exists z_0∈D such that R(z)-α(z_0):=(z-β_0)~pH(z) has at most two distinct zeros(or poles)β_0 and suppose that the multiplicities l and k of zeros of f(z)-β_0 andα(z)-α(z_0) at z_0,respectively,satisfy k≠l|p|,where H(z) is a rational function and H(β_0)≠0,∞andα(z_0)∈C U {∞} andα(z) is nonconstant.Then F is normal in D.In particular,the result is a kind of generalization of the famous Montel's criterion.

Direct and inverse theorems of rational approximation in the Bergman space

For positive numbers and let denote the Bergman space of analytic functions in the half-plane . For let be the best approximation by rational functions of degree at most . Also let and be numbers such that and . Then the main result of the paper claims that the set of functions such that is precisely the Besov space of analytic functions in .Bibliography: 23 titles.

Ramification in iterated towers for rational functions

Let \({\phi(x)}\) be a rational function of degree > 1 defined over a number field K and let \({\Phi_{n}(x,t) = \phi^{(n)}(x)-t \in K(x,t)}\) where \({\phi^{(n)}(x)}\) is the nth iterate of \({\phi(x)}\). We give a formula for the discriminant of the numerator of Φn(x, t) and show that, if \({\phi(x)}\) is postcritically finite, for each specialization t0 of t to K, there exists a finite set \({S_{t_0}}\) of primes of K such that for all n, the primes dividing the discriminant are contained in \({S_{t_0}}\).

A quantitative estimate for quasiintegral points in orbits

Let ϕ(z) ∈ K(z) be a rational function of degree d ≥ 2 defined over a number field whose second iterate ϕ 2 (z) is not a polynomial, and let α ∈ K. The second author previously proved that the forward orbit O ϕ (α) contains only finitely many quasi-S-integral points. We give an explicit upper bound for the number of such points.

SEMIGROUP OF RATIONAL p-ADIC FUNCTIONS FOR COMPOSITION

We are interested in the Julia set of a semigroup of rational functions with coefficients in $\C_p$ where the semigroup operation is composition. We prove that if a semigroup $G$ is generated by a finite number of rational functions of degree at least two with coefficients in a finite extension of $\Q_p$, and has a nonempty Julia set $\cj(G)$, then $\cj(G)$ is perfect and has an empty interior.

About the p-adic Yosida equation inside a disk

Let K be a complete ultrametric algebraically closed field and let ℳ( d(0, R ℒ)) be the field of meromorphic functions inside the disk d(0, R −) = { x ∈ K ∣ ∣x∣ < R}. Let ℳ b ( d(0, R ℒ)) be the subfield of bounded meromorphic functions inside d(0, R −) and let ℳ u ( d(0, R ℒ)) = ℳ( d(0, R ℒ)) ∖ ℳ b ( d(0, R ℒ)) be the subset of unbounded meromorphic functions inside d(0, R −). Initially, we consider the Yosida Equation: ( d y d x ) m = F ( y ) , where m ∈ ℕ * and F(X) is a rational function of degree d with coefficients in ℳ b ( d(0, R ℒ)). We show that, if d ≥ 2 m + 1, this equation has no solution in ℳ u ( d(0, R ℒ)). Next, we examine solutions of the above equation when F(X) is apolynomial with constant coefficients and show that it has no unbounded analytic functions in d(0, R −). Further, we list the only cases when the equation may eventually admit solutions in ℳ u ( d(0, R ℒ)). Particularly, the elliptic equation may not.

RATIONAL MAPS ADMITTING MEROMORPHIC INVARIANT LINE FIELDS

AbstractIt is shown that a rational map of degree at least 2 admits a meromorphic invariant line field if and only if it is conformally conjugate to either an integral Lattès map, a power map, or a Chebyshev polynomial.

Primitive divisors in arithmetic dynamics

AbstractLet ϕ(z) ∈ (z) be a rational function of degree d ≥ 2 with ϕ(0) = 0 and such that ϕ does not vanish to order d at 0. Let α ∈ have infinite orbit under iteration of ϕ and write ϕn(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and p ∤ Ai for all i &lt; n. More generally, we prove an analogous result when ϕ is defined over a number field and 0 is a preperiodic point for ϕ.

A finiteness theorem for canonical heights attached to rational maps over function fields

Let K be a function field, let φ ∈ K(T ) be a rational map of degree d ≥ 2 defined over K, and suppose that φ is not isotrivial. In this paper, we show that a point P ∈ P(K) has φ-canonical height zero if and only if P is preperiodic for φ. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually prove the following stronger result, which is a variant of the Northcott finiteness principle: there exists e > 0 such that the set of points P ∈ P(K) with φ-canonical height at most e is finite. Our proof is essentially analytic, making use of potential theory on Berkovich spaces to study the dynamical Green’s functions gφ,v(x, y) attached to φ at each place v of K. For example, we show that every conjugate of φ has bad reduction at v if and only if gφ,v(x, x) > 0 for all x ∈ P1Berk,v, where P 1 Berk,v denotes the Berkovich projective line over the completion of Kv. In an appendix, we use a similar method to give a new proof of the Mordell-Weil theorem for elliptic curves over K.

A local-global criterion for dynamics on P1

Let K be a number field or a function field, let F:P^1 --> P^1 be a rational map of degree at least two defined over K, let P be a point in P^1(K) having infinite F-orbit, and let Z be a finite subset of Z. We prove a local-global criterion for the intersection of the F-orbit of P and the finite set Z. This is a special case of a dynamical Brauer-Manin criterion suggested by Hsia and Silverman.