arXiv subjects

Benjamin Hutz

Publications and source records attributed to Benjamin Hutz.

28 records · Page 2Linked to original sources

arXiv2011

Pre-images of quadratic dynamical systems

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 a given constant after iteration, called pre-images of the constant. In the article "Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems," it was shown that the number of rational pre-images is bounded as one varies the morphism in a certain one-dimensional family. Explicit values of the constant for pre-images of zero and -1 defined over the rational numbers were addressed in subsequent articles. This article addresses an explicit bound for any algebraic image constant and provides insight into the geometry of the "pre-image surfaces."

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arXiv2011

Almost Newton, sometimes Lattès

Self-maps everywhere defined on the projective space $¶^N$ over a number field or a function field are the basic objects of study in the arithmetic of dynamical systems. One reason is a theorem of Fakkruddin \cite{Fakhruddin} (with complements in \cite{Bhatnagar}) that asserts that a "polarized" self-map of a projective variety is essentially the restriction of a self-map of the projective space given by the polarization. In this paper we study the natural self-maps defined the following way: $F$ is a homogeneous polynomial of degree $d$ in $(N+1)$ variables $X_i$ defining a smooth hypersurface. Suppose the characteristic of the field does not divide $d$ and define the map of partial derivatives $ϕ_F = (F_{X_0},...,F_{X_N})$. The map $ϕ_F$ is defined everywhere due to the following formula of Euler: $\sum X_i F_{X_i} = d F$, which implies that a point where all the partial derivatives vanish is a non-smooth point of the hypersuface F=0. One can also compose such a map with an element of $\PGL_{N+1}$. In the particular case addressed in this article, N=1, the smoothness condition means that $F$ has only simple zeroes. In this manner, fixed points and their multipliers are easy to describe and, moreover, with a few modifications we recover classical dynamical systems like the Newton method for finding roots of polynomials or the Lattès map corresponding to the multiplication by 2 on an elliptic curve.

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arXiv2011

Effectivity of Dynatomic cycles for morphisms of projective varieties using deformation theory

Given an endomorphism of a projective variety, by intersecting the graph and the diagonal varieties we can determine the set of periodic points. In an effort to determine the periodic points of a given minimal period, we follow a construction similar to cyclotomic polynomials. The resulting zero-cycle is called a dynatomic cycle and the points in its support are called formal periodic points. This article gives a proof of the effectivity of dynatomic cycles for morphisms of projective varieties using methods from deformation theory.

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arXiv2010

On the Number of Rational Iterated Pre-images of the Origin Under Quadratic Dynamical Systems

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.''

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arXiv2010

On Poonen's Conjecture Concerning Rational Preperiodic Points of Quadratic Maps

The purpose of this note is give some evidence in support of conjectures of Poonen, and Morton and Silverman, on the periods of rational numbers under the iteration of quadratic polynomials. In particular, Poonen conjectured that there are at most 9 periodic points defined over the rational numbers for any map in the family x^2 + c for c rational. We verify this conjecture for c values up to height 10^8. For quadratic number fields, we provide evidence that the upper bound on the exact period of Q-rational periodic point is 6.

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arXiv2010

Good Reduction of Periodic Points

We consider the dynamical system created by iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. We study the primitive period of a periodic point in this field in relation to the primitive period of the reduced point in the residue field, the order of the action on the cotangent space, and the characteristic of the residue field.

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arXiv2010

Finding Rational Periodic Points on Wehler K3 Surfaces

This article examines dynamical systems on a class of K3 surfaces in $\mathbb{P}^{2} \times \mathbb{P}^{2}$ with an infinite automorphism group. In particular, this article develops an algorithm to find $\mathbb{Q}$-rational periodic points using information modulo $p$ for various primes $p$. The algorithm is applied to exhibit K3 surfaces with $\mathbb{Q}$-rational periodic points of primitive period $1,...,16$. A portion of the algorithm is then used to determine the Riemann zeta function modulo 3 of a particular K3 surface and find a family of K3 surfaces with Picard number two.

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arXiv2009

Rational Periodic Points for Degree Two Polynomial Morphisms on Projective Space

This article addresses the existence of $\Q$-rational periodic points for morphisms of projective space. In particular, we construct an infinitely family of morphisms on $¶^N$ where each component is a degree 2 homogeneous form in $N+1$ variables which has a $\Q$-periodic point of primitive period $\frac{(N+1)(N+2)}{2} + \lfloor \frac{N-1}{2}\rfloor$. This result is then used to show that for $N$ large enough there exists morphisms of $¶^N$ with $\Q$-rational periodic points with primitive period larger that $c(k)N^k$ for any $k$ and some constant $c(k)$.

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arXiv2008

Dynatomic cycles for morphisms of projective varieties

We prove the effectivity of the dynatomic cycles for morphisms of projective varieties. We then analyze the degrees of the dynatomic cycles and multiplicities of formal periodic points and apply these results to the existence of periodic points with arbitrarily large primitive periods.

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arXiv2008

Uniform Bounds on Pre-Images under Quadratic Dynamical Systems

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.

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