Preperiodic points for rational functions defined over a rational function field of characteristic zero

Let $k$ be an algebraic closed field of characteristic zero. Let $K$ be the rational function field $K=k(t)$. Let $\phi$ be a non isotrivial rational function in $K(z)$. We prove a bound for the cardinality of the set of $K$--rational preperiodic points for $\phi$ in terms of the number of places of bad reduction and the degree $d$ of $\phi$.Comment: There was a mistake in the previous version. The results have been proven only for rational function field

Good Reduction for Endomorphisms of the Projective Line in Terms of the Branch Locus

Let $K$ be a number field and $v$ a non archimedean valuation on $K$. We say that an endomorphism $\Phi\colon \mathbb{P}_1\to \mathbb{P}_1$ has good reduction at $v$ if there exists a model $\Psi$ for $\Phi$ such that $\deg\Psi_v$, the degree of the reduction of $\Psi$ modulo $v$, equals $\deg\Psi$ and $\Psi_v$ is separable. We prove a criterion for good reduction that is the natural generalization of a result due to Zannier in \cite{Uz3}. Our result is in connection with other two notions of good reduction, the simple and the critically good reduction. The last part of our article is dedicated to prove a characterization of the maps whose iterates, in a certain sense, preserve the critically good reduction.Comment: 23 pages, comments are welcom

Quadratic maps with a periodic critical point of period 2

We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 7. We explain how this extends results of Poonen on quadratic polynomials. We show that there are 13 possible graphs, and that such maps have at most 9 rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic critical point of period 3 or 4.Comment: Updated theorem 2 to rule out the cases of quadratic maps with a rational periodic critical point of period 2 and a rational periodic point of period 5 or

Cycles for rational maps with good reduction outside a prescribed set

Let $K$ be a number field and $S$ a fixed finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we study the cycles for rational maps of $\mathbb{P}_1(K)$ of degree $\geq2$ with good reduction outside $S$. We say that two ordered $n$-tuples $(P_0,P_1,...,P_{n-1})$ and $(Q_0,Q_1,...,Q_{n-1})$ of points of $\mathbb{P}_1(K)$ are equivalent if there exists an automorphism $A\in{\rm PGL}_2(R_S)$ such that $P_i=A(Q_i)$ for every index $i\in\{0,1,...,n-1\}$. We prove that if we fix two points $P_0,P_1\in\mathbb{P}_1(K)$, then the number of inequivalent cycles for rational maps of degree $\geq2$ with good reduction outside $S$ which admit $P_0,P_1$ as consecutive points is finite and depends only on $S$. We also prove that this result is in a sense best possible.Comment: 30 pages, changed conten

Rational periodic points for quadratic maps

Let $K$ be a number field. Let $S$ be a finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we consider endomorphisms of \pro of degree 2, defined over $K$, with good reduction outside $S$. We prove that there exist only finitely many such endomorphisms, up to conjugation by ${\rm PGL}_2(R_S)$, admitting a periodic point in \po of order $>3$. Also, all but finitely many classes with a periodic point in \po of order 3 are parametrized by an irreducible curve.Comment: 32 pages. To appear on Annales de l'Insitut Fourier. Corrected some mistakes in the proofs of Lemma 6 and Lemma 8. Thanks to the refere

Precise 3D track reconstruction algorithm for the ICARUS T600 liquid argon time projection chamber detector

Liquid Argon Time Projection Chamber (LAr TPC) detectors offer charged particle imaging capability with remarkable spatial resolution. Precise event reconstruction procedures are critical in order to fully exploit the potential of this technology. In this paper we present a new, general approach of three-dimensional reconstruction for the LAr TPC with a practical application to track reconstruction. The efficiency of the method is evaluated on a sample of simulated tracks. We present also the application of the method to the analysis of real data tracks collected during the ICARUS T600 detector operation with the CNGS neutrino beam.Comment: Submitted to Advances in High Energy Physic

Search for anomalies in the {\nu}e appearance from a {\nu}{\mu} beam

We report an updated result from the ICARUS experiment on the search for {\nu}{\mu} ->{\nu}e anomalies with the CNGS beam, produced at CERN with an average energy of 20 GeV and travelling 730 km to the Gran Sasso Laboratory. The present analysis is based on a total sample of 1995 events of CNGS neutrino interactions, which corresponds to an almost doubled sample with respect to the previously published result. Four clear {\nu}e events have been visually identified over the full sample, compared with an expectation of 6.4 +- 0.9 events from conventional sources. The result is compatible with the absence of additional anomalous contributions. At 90% and 99% confidence levels the limits to possible oscillated events are 3.7 and 8.3 respectively. The corresponding limit to oscillation probability becomes consequently 3.4 x 10-3 and 7.6 x 10-3 respectively. The present result confirms, with an improved sensitivity, the early result already published by the ICARUS collaboration