arXiv subjects
Publications and source records attributed to Xander Faber.
We refine a result of R. Benedetto in p-adic analysis in order to exhibit infinitely many quadratic polynomials with rational coefficients having a specified graph of rational preperiodic points.
Given a nonconstant holomorphic map f: X -> Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor R_f is a finite formal linear combination of points of X that is combinatorially constrained by the Hurwitz formula. Now let k be an algebraically closed field that is complete with respect to a nontrivial non-Archimedean absolute value. For example, k = C_p. Here the role of a Riemann surface is played by a projective Berkovich analytic curve. As these curves have many points that are not algebraic over k, some new (non-algebraic) ramification behavior appears for maps between them. For example, the ramification locus is no longer a divisor, but rather a closed analytic subspace. This article initiates a detailed study of the ramification locus for self-maps f: P^1 -> P^1. This simplest first case has the benefit of being approachable by concrete (and often combinatorial) techniques.
This article is the second installment in a series on the Berkovich ramification locus for nonconstant rational functions f: P^1 -> P^1. Here we show the ramification locus of f is contained in a strong tubular neighborhood of finite radius around the connected hull of the critical points if and only if f is tamely ramified at all of its critical points. When the ground field has characteristic zero, this bound may be chosen to depend only on the residue characteristic. We give two applications to classical non-Archimedean analysis, including a new version of Rolle's theorem for rational functions.
Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any field K equipped with a set of pairwise inequivalent absolute values satisfying a product formula, we completely describe the conditions under which Newton's method applied to a squarefree polynomial f with K-coefficients will succeed in finding a root of f in the v-adic topology for infinitely many places v of K. Furthermore, we show that if K is a finite extension of the rationals or of the rational function field over a finite field, then the Newton approximation sequence fails to converge v-adically for a positive density of places v.
Let phi and psi be endomorphisms of the projective line of degree at least 2, defined over a noetherian commutative ring R with unity. From a dynamical perspective, a significant question is to determine whether phi and psi are conjugate (or to answer the related question of whether a given map phi has a nontrivial automorphism). We show that the space of automorphisms of P^1 conjugating phi to psi is a finite subscheme of PGL(2) (respectively that the automorphism group of phi is a finite group scheme). We construct efficient algorithms for computing the set of conjugating maps (resp. the group of automorphisms) when R is a field. Each of our algorithms takes advantage of different dynamical structures, so context (e.g., field of definition and degree of the map) determines the preferred algorithm. We have implemented them in Sage when R is a finite field or the field of rational numbers, and we give running times for computing automorphism groups for hundreds of random endomorphisms of P^1. These examples demonstrate the superiority of these new algorithms over a naive approach using Groebner bases.
Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference x_{n+1}-x_n has a primitive prime factor for all sufficiently large n. This result provides a new proof of the infinitude of primes for each rational function f of degree at least 2.
Over an algebraically closed field of positive characteristic, there exist rational functions with only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we use this tool to discern some of the basic geometry of the space of unicritical rational functions, as well as its quotients by the SL(2)-actions of conjugation and postcomposition. We also give an application to dynamical systems with restricted ramification defined over non-Archimedean fields of positive residue characteristic.
Let X be a tropical curve (or metric graph), and fix a base point p on X. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(X) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for X. This result is useful for reducing certain questions about the Abel-Jacobi map Phi_p : X -> J(X), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over the rationals. As an application of our direct limit theorem, we derive some local comparison formulas between g and its pullback Phi_p^*(g) for three different natural "metrics" g on J(X). One of these formulas implies that Phi_p is a tropical isometry when X is 2-edge-connected. Another shows that the canonical measure on a metric graph X, defined by S. Zhang, measures lengths on the image Phi_p(X) with respect to the "sup-norm" on J(X).
Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x_0 be a sufficiently general element of K, and let alpha be a root of f. We give precise conditions under which Newton iteration, started at the point x_0, converges v-adically to the root alpha for infinitely many places v of K. As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v-adically to any given root of f for infinitely many places v. We also conjecture that the set of places for which Newton iteration diverges has full density and give some heuristic and numerical evidence.
For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article ``Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems,'' by two of the present authors and five others, it was shown that the number of rational iterated pre-images of the origin is bounded as one varies the morphism in a certain one-dimensional family. Subject to the validity of the Birch and Swinnerton-Dyer conjecture and some other related conjectures for the L-series of a specific abelian variety and using a number of modern tools for locating rational points on high genus curves, we show that the maximum number of rational iterated pre-images is six. We also provide further insight into the geometry of the ``pre-image curves.''
For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper bound on the number of elements of G(b,c) whose degree over K is at most some constant B. This bound depends only on b, [K:Q], and B, and is valid for all b outside an explicit finite set. We also show that, for any N>3 and any b in K outside a finite set, there are only finitely many pairs of complex numbers (y,c) for which [K(y,c):K]<2^(N-3) and the value of the N-th iterate of f_c(x) at x=y is b. Moreover, the bound 2^(N-3) in this result is optimal.
A metrized graph is a finite weighted graph whose edges are thought of as line segments. In this expository paper, we study the Laplacian operator on a metrized graph and some important functions related to it, including the ``j-function'', the effective resistance, and eigenfunctions of the Laplacian. We discuss the relationship between metrized graphs and electrical networks, which provides some physical intuition for the concepts being dealt with. We also discuss the relation between the Laplacian on a metrized graph and the combinatorial Laplacian matrix. We introduce the``canonical measure'' on a metrized graph, which arises naturally when considering the Laplacian of the effective resistance function. Finally, we discuss a generalization of classical Fourier analysis which utilizes eigenfunctions of the Laplacian on a metrized graph. During the course of the paper, we obtain a proof of Foster's network theorem and of an intriguing series identity.