7,303 research outputs found
Factoring bivariate lacunary polynomials without heights
We present an algorithm which computes the multilinear factors of bivariate
lacunary polynomials. It is based on a new Gap Theorem which allows to test
whether a polynomial of the form P(X,X+1) is identically zero in time
polynomial in the number of terms of P(X,Y). The algorithm we obtain is more
elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on
the valuation of polynomials of the previous form instead of the height of the
coefficients. As a result, it can be used to find some linear factors of
bivariate lacunary polynomials over a field of large finite characteristic in
probabilistic polynomial time.Comment: 25 pages, 1 appendi
Perfect single error-correcting codes in the Johnson Scheme
Delsarte conjectured in 1973 that there are no nontrivial pefect codes in the
Johnson scheme. Etzion and Schwartz recently showed that perfect codes must be
k-regular for large k, and used this to show that there are no perfect codes
correcting single errors in J(n,w) for n <= 50000. In this paper we show that
there are no perfect single error-correcting codes for n <= 2^250.Comment: 4 pages, revised, accepted for publication in IEEE Transactions on
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Integer Factorization with a Neuromorphic Sieve
The bound to factor large integers is dominated by the computational effort
to discover numbers that are smooth, typically performed by sieving a
polynomial sequence. On a von Neumann architecture, sieving has log-log
amortized time complexity to check each value for smoothness. This work
presents a neuromorphic sieve that achieves a constant time check for
smoothness by exploiting two characteristic properties of neuromorphic
architectures: constant time synaptic integration and massively parallel
computation. The approach is validated by modifying msieve, one of the fastest
publicly available integer factorization implementations, to use the IBM
Neurosynaptic System (NS1e) as a coprocessor for the sieving stage.Comment: Fixed typos in equation for modular roots (Section II, par. 6;
Section III, par. 2) and phase calculation (Section IV, par 2
On the difficulty of presenting finitely presentable groups
We exhibit classes of groups in which the word problem is uniformly solvable
but in which there is no algorithm that can compute finite presentations for
finitely presentable subgroups. Direct products of hyperbolic groups, groups of
integer matrices, and right-angled Coxeter groups form such classes. We discuss
related classes of groups in which there does exist an algorithm to compute
finite presentations for finitely presentable subgroups. We also construct a
finitely presented group that has a polynomial Dehn function but in which there
is no algorithm to compute the first Betti number of the finitely presentable
subgroups.Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewal
Notes on the multiplicity conjecture
New cases of the multiplicity conjecture are considered
Arithmetic properties of blocks of consecutive integers
This paper provides a survey of results on the greatest prime factor, the
number of distinct prime factors, the greatest squarefree factor and the
greatest m-th powerfree part of a block of consecutive integers, both without
any assumption and under assumption of the abc-conjecture. Finally we prove
that the explicit abc-conjecture implies the Erd\H{o}s-Woods conjecture for
each k>2.Comment: A slightly corrected and extended version of a paper which will
appear in January 2017 in the book From Arithmetic to Zeta-functions
published by Springe
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