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A criterion for potentially good reduction in nonarchimedean dynamics

Robert L. Benedetto (2014)

Acta Arithmetica

Let K be a nonarchimedean field, and let ϕ ∈ K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sufficient condition, involving only the fixed points of ϕ and their preimages, that determines whether or not the dynamical system ϕ: ℙ¹ → ℙ¹ has potentially good reduction.

Bounds on the radius of the p-adic Mandelbrot set

Jacqueline Anderson (2013)

Acta Arithmetica

Let f ( z ) = z d + a d - 1 z d - 1 + . . . + a 1 z p [ z ] be a degree d polynomial. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p ≥ d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for 1/2d ≤ p < d. We also explore a one-parameter family of cubic polynomials over ℚ₂ to illustrate that the p-adic Mandelbrot set can be quite complicated when p < d, in contrast with the simple and...

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

We consider a rational function f which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number of terms. Then we look at the possible decompositions f ( x ) = g ( h ( x ) ) , where g , h are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of g is bounded only in terms of (and we provide explicit bounds). This supports and quantifies the intuitive...

Dynamique des polynômes quadratiques sur les corps locaux

Robert Benedetto, Jean-Yves Briend, Hervé Perdry (2007)

Journal de Théorie des Nombres de Bordeaux

Dans cette note, nous montrons que la dynamique d’un polynôme quadratique sur un corps local peut être déterminée en temps fini, et que l’on a l’alternative suivante : soit l’ensemble de Julia est vide, soit P y est conjugué au décalage unilatéral sur 2 symboles.

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

Tatiana Bandman, Shelly Garion, Boris Kunyavskiĭ (2014)

Open Mathematics

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

Equidistribution of preimages over nonarchimedean fields for maps of good reduction

William Gignac (2014)

Annales de l’institut Fourier

In this article we prove an analogue of the equidistribution of preimages theorem from complex dynamics for maps of good reduction over nonarchimedean fields. While in general our result is only a partial analogue of the complex equidistribution theorem, for most maps of good reduction it is a complete analogue. In the particular case when the nonarchimedean field in question is equipped with the trivial absolute value, we are able to supply a strengthening of the theorem, namely that the preimages...

Newton’s method over global height fields

Xander Faber, Adam Towsley (2014)

Journal de Théorie des Nombres de Bordeaux

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 K x will succeed in finding some 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...

On some issues concerning polynomial cycles

Tadeusz Pezda (2013)

Communications in Mathematics

We consider two issues concerning polynomial cycles. Namely, for a discrete valuation domain R of positive characteristic (for N 1 ) or for any Dedekind domain R of positive characteristic (but only for N 2 ), we give a closed formula for a set 𝒞 Y C L ( R , N ) of all possible cycle-lengths for polynomial mappings in R N . Then we give a new property of sets 𝒞 Y C L ( R , 1 ) , which refutes a kind of conjecture posed by W. Narkiewicz.

On the number of places of convergence for Newton’s method over number fields

Xander Faber, José Felipe Voloch (2011)

Journal de Théorie des Nombres de Bordeaux

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 α 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 α 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...

Polynomial cycles in certain local domains

T. Pezda (1994)

Acta Arithmetica

1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple x , x , . . . , x k - 1 of distinct elements of R is called a cycle of f if f ( x i ) = x i + 1 for i=0,1,...,k-2 and f ( x k - 1 ) = x . The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number 7 7 · 2 N , depending only on the degree N of K. In this note we consider...

The arithmetic of curves defined by iteration

Wade Hindes (2015)

Acta Arithmetica

We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...

The minimal resultant locus

Robert Rumely (2015)

Acta Arithmetica

Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map γ o r d ( R e s ( φ γ ) ) factors through a function o r d R e s φ ( · ) on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in P ¹ K , or on a segment, and the minimal resultant locus is contained in the tree in P ¹ K spanned by the fixed points and poles...

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