4,264 research outputs found
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
Splitting full matrix algebras over algebraic number fields
Let K be an algebraic number field of degree d and discriminant D over Q. Let
A be an associative algebra over K given by structure constants such that A is
isomorphic to the algebra M_n(K) of n by n matrices over K for some positive
integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with
M_n(K) can be constructed by a polynomial time ff-algorithm. (An ff-algorithm
is a deterministic procedure which is allowed to call oracles for factoring
integers and factoring univariate polynomials over finite fields.)
As a consequence, we obtain a polynomial time ff-algorithm to compute
isomorphisms of central simple algebras of bounded degree over K.Comment: 15 pages; Theorem 2 and Lemma 8 correcte
New Einstein-Sasaki and Einstein Spaces from Kerr-de Sitter
In this paper, which is an elaboration of our results in hep-th/0504225, we
construct new Einstein-Sasaki spaces L^{p,q,r_1,...,r_{n-1}} in all odd
dimensions D=2n+1\ge 5. They arise by taking certain BPS limits of the
Euclideanised Kerr-de Sitter metrics. This yields local Einstein-Sasaki metrics
of cohomogeneity n, with toric U(1)^{n+1} principal orbits, and n real
non-trivial parameters. By studying the structure of the degenerate orbits we
show that for appropriate choices of the parameters, characterised by the (n+1)
coprime integers (p,q,r_1,...,r_{n-1}), the local metrics extend smoothly onto
complete and non-singular compact Einstein-Sasaki manifolds
L^{p,q,r_1,...,r_{n-1}}. We also construct new complete and non-singular
compact Einstein spaces \Lambda^{p,q,r_1,...,r_n} in D=2n+1 that are not
Sasakian, by choosing parameters appropriately in the Euclideanised Kerr-de
Sitter metrics when no BPS limit is taken.Comment: latex, 26 page
The Road to Quantum Computational Supremacy
We present an idiosyncratic view of the race for quantum computational
supremacy. Google's approach and IBM challenge are examined. An unexpected
side-effect of the race is the significant progress in designing fast classical
algorithms. Quantum supremacy, if achieved, won't make classical computing
obsolete.Comment: 15 pages, 1 figur
Linear independence of powers of singular moduli of degree 3
We show that two distinct singular moduli , such that for
some positive integers the numbers and are
linearly dependent over generate the same number field of degree
at most . This completes a result of Riffaut, who proved the above theorem
except for two explicit pair of exceptions consisting of numbers of degree .
The purpose of this article is to treat these two remaining cases
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