25,812 research outputs found
Curvature on the integers, I
Starting with a symmetric/antisymmetric matrix with integer coefficients
(which we view as an analogue of a metric/form on a principal bundle over the
"manifold" Spec Z) we introduce arithmetic analogues of Chern connections and
their curvature (in which usual partial derivative operators acting on
functions are replaced by Fermat quotient operators acting on integer numbers);
curvature is introduced via the method of "analytic continuation between
primes" \cite{laplace}. We prove various non-vanishing, respectively vanishing
results for curvature; morally, Spec Z will appear as "intrinsically curved."
Along with \cite{adel1, adel2, adel3}, this theory can be viewed as taking
first steps in developing a "differential geometry of Spec Z.
Ring-LWE Cryptography for the Number Theorist
In this paper, we survey the status of attacks on the ring and polynomial
learning with errors problems (RLWE and PLWE). Recent work on the security of
these problems [Eisentr\"ager-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives
rise to interesting questions about number fields. We extend these attacks and
survey related open problems in number theory, including spectral distortion of
an algebraic number and its relationship to Mahler measure, the monogenic
property for the ring of integers of a number field, and the size of elements
of small order modulo q.Comment: 20 Page
Celebrating President Farish: A Life and Legacy
On Wednesday, the Roger Williams University community and greater community gathered to share a powerful celebration of the life and legacy of Donald J. Farish, who died on July 5, 2018, while serving his eighth year as our president
Factoring bivariate sparse (lacunary) polynomials
We present a deterministic algorithm for computing all irreducible factors of
degree of a given bivariate polynomial over an algebraic
number field and their multiplicities, whose running time is polynomial in
the bit length of the sparse encoding of the input and in . Moreover, we
show that the factors over \Qbarra of degree which are not binomials
can also be computed in time polynomial in the sparse length of the input and
in .Comment: 20 pp, Latex 2e. We learned on January 23th, 2006, that a
multivariate version of Theorem 1 had independently been achieved by Erich
Kaltofen and Pascal Koira
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
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