191,138 research outputs found
Some Aspects and Applications of the Riemann Hypothesis over Finite Fields
We give a survey of some aspects of the Riemann Hypothesis over finite fields, as it was proved by Deligne, and its applications to analytic number theory. In particular, we concentrate on the formalism leading to Deligne's Equidistribution Theore
Locally finite graphs with ends: A topological approach, I. Basic theory
AbstractThis paper is the first of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [18]. They are complemented by a third part [28], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the second part of this survey [19] will concentrate on those applications, this first part explores the new theory as such: it introduces the basic concepts and facts, describes some of the proof techniques that have emerged over the past 10 years (as well as some of the pitfalls these proofs have in stall for the naive explorer), and establishes connections to neighbouring fields such as algebraic topology and infinite matroids. Numerous open problems are suggested
Recognising Multidimensional Euclidean Preferences
Euclidean preferences are a widely studied preference model, in which
decision makers and alternatives are embedded in d-dimensional Euclidean space.
Decision makers prefer those alternatives closer to them. This model, also
known as multidimensional unfolding, has applications in economics,
psychometrics, marketing, and many other fields. We study the problem of
deciding whether a given preference profile is d-Euclidean. For the
one-dimensional case, polynomial-time algorithms are known. We show that, in
contrast, for every other fixed dimension d > 1, the recognition problem is
equivalent to the existential theory of the reals (ETR), and so in particular
NP-hard. We further show that some Euclidean preference profiles require
exponentially many bits in order to specify any Euclidean embedding, and prove
that the domain of d-Euclidean preferences does not admit a finite forbidden
minor characterisation for any d > 1. We also study dichotomous preferencesand
the behaviour of other metrics, and survey a variety of related work.Comment: 17 page
Generic Newton polygons for curves of given p-rank
We survey results and open questions about the -ranks and Newton polygons
of Jacobians of curves in positive characteristic . We prove some geometric
results about the -rank stratification of the moduli space of
(hyperelliptic) curves. For example, if , we prove that
every component of the -rank stratum of contains a
component of the -rank stratum in its closure. We prove that the
-rank stratum of is connected. For all primes
and all , we demonstrate the existence of a Jacobian of a smooth
curve, defined over , whose Newton polygon has slopes
. We include partial results about the
generic Newton polygons of curves of given genus and -rank .Comment: 16 pages, to appear in Algebraic Curves and Finite Fields: Codes,
Cryptography, and other emergent applications, edited by H. Niederreiter, A.
Ostafe, D. Panario, and A. Winterho
Notions of Tensor Rank
Tensors, or multi-linear forms, are important objects in a variety of areas
from analytics, to combinatorics, to computational complexity theory. Notions
of tensor rank aim to quantify the "complexity" of these forms, and are thus
also important. While there is one single definition of rank that completely
captures the complexity of matrices (and thus linear transformations), there is
no definitive analog for tensors. Rather, many notions of tensor rank have been
defined over the years, each with their own set of uses. In this paper we
survey the popular notions of tensor rank. We give a brief history of their
introduction, motivating their existence, and discuss some of their
applications in computer science. We also give proof sketches of recent results
by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between
three key notions of tensor rank over finite fields with at least three
elements
On Random Sampling of Supersingular Elliptic Curves
We consider the problem of sampling random supersingular elliptic curves over finite fields of cryptographic size (SRS problem). The currently best-known method combines the reduction of a suitable complex multiplication (CM) -invariant and a random walk over some supersingular isogeny graph. Unfortunately, this method is not suitable for numerous cryptographic applications because it gives information about the endomorphism ring of the generated curve. This motivates a stricter version of the SRS problem, requiring that the sampling algorithm gives no information about the endomorphism ring of the output curve (cSRS problem).
In this work we formally define the SRS and cSRS problems, which both enjoy a theoretical interest. We discuss the relevance of the latter also for cryptographic applications, and we provide a self-contained survey of the known approaches to both problems. Those for the cSRS problem work only for small finite fields, have exponential complexity in the characteristic of the base finite field (since they require computing and finding roots of polynomials of large degree), leaving the problem open. In the second part of the paper, we propose and analyse some alternative techniques — based either on Hasse invariant or division polynomials — and we explain the reasons why them do not readily lead to efficient cSRS algorithms, but they may open promising research directions
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