59,276 research outputs found
Nowhere Dense Graph Classes and Dimension
Nowhere dense graph classes provide one of the least restrictive notions of
sparsity for graphs. Several equivalent characterizations of nowhere dense
classes have been obtained over the years, using a wide range of combinatorial
objects. In this paper we establish a new characterization of nowhere dense
classes, in terms of poset dimension: A monotone graph class is nowhere dense
if and only if for every and every , posets of height
at most with elements and whose cover graphs are in the class have
dimension .Comment: v4: Minor changes suggested by a refere
Families of abelian varieties with many isogenous fibres
Let Z be a subvariety of the moduli space of principally polarised abelian
varieties of dimension g over the complex numbers. Suppose that Z contains a
Zariski dense set of points which correspond to abelian varieties from a single
isogeny class. A generalisation of a conjecture of Andr\'e and Pink predicts
that Z is a weakly special subvariety. We prove this when dim Z = 1 using the
Pila--Zannier method and the Masser--W\"ustholz isogeny theorem. This
generalises results of Edixhoven and Yafaev when the Hecke orbit consists of CM
points and of Pink when it consists of Galois generic points.Comment: Gap in Lemma 3.3 found and corrected by Gabriel Dil
O-minimality and certain atypical intersections
We show that the strategy of point counting in o-minimal structures can be
applied to various problems on unlikely intersections that go beyond the
conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called
Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound
for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In
the context of abelian varieties we obtain the Zilber-Pink Conjecture for
curves unconditionally when everything is defined over a number field. For
higher dimensional subvarieties of abelian varieties we obtain some weaker
results and some conditional results
Querying the Guarded Fragment
Evaluating a Boolean conjunctive query Q against a guarded first-order theory
F is equivalent to checking whether "F and not Q" is unsatisfiable. This
problem is relevant to the areas of database theory and description logic.
Since Q may not be guarded, well known results about the decidability,
complexity, and finite-model property of the guarded fragment do not obviously
carry over to conjunctive query answering over guarded theories, and had been
left open in general. By investigating finite guarded bisimilar covers of
hypergraphs and relational structures, and by substantially generalising
Rosati's finite chase, we prove for guarded theories F and (unions of)
conjunctive queries Q that (i) Q is true in each model of F iff Q is true in
each finite model of F and (ii) determining whether F implies Q is
2EXPTIME-complete. We further show the following results: (iii) the existence
of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof
of the finite model property of the clique-guarded fragment; (v) the small
model property of the guarded fragment with optimal bounds; (vi) a
polynomial-time solution to the canonisation problem modulo guarded
bisimulation, which yields (vii) a capturing result for guarded bisimulation
invariant PTIME.Comment: This is an improved and extended version of the paper of the same
title presented at LICS 201
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