235 research outputs found

### 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

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