5,322 research outputs found
Differences of Harmonic Numbers and the -Conjecture
Our main source of inspiration was a talk by Hendrik Lenstra on harmonic
numbers, which are numbers whose only prime factors are two or three.
Gersonides proved 675 years ago that one can be written as a difference of
harmonic numbers in only four ways: 2-1, 3-2, 4-3, and 9-8. We investigate
which numbers other than one can or cannot be written as a difference of
harmonic numbers and we look at their connection to the -conjecture. We
find that there are only eleven numbers less than 100 that cannot be written as
a difference of harmonic numbers (we call these -numbers). The smallest
-number is 41, which is also Euler's largest lucky number and is a very
interesting number. We then show there are infinitely many -numbers, some
of which are the primes congruent to modulo . For each Fermat or
Mersenne prime we either prove that it is an -number or find all ways it
can be written as a difference of harmonic numbers. Finally, as suggested by
Lenstra in his talk, we interpret Gersonides' theorem as "The -conjecture
is true on the set of harmonic numbers" and we expand the set on which the
-conjecture is true by adding to the set of harmonic numbers the following
sets (one at a time): a finite set of -numbers, the infinite set of primes
of the form , the set of Fermat primes, and the set of Mersenne primes.Comment: 13 pages, 1 figur
Parallelization of Modular Algorithms
In this paper we investigate the parallelization of two modular algorithms.
In fact, we consider the modular computation of Gr\"obner bases (resp. standard
bases) and the modular computation of the associated primes of a
zero-dimensional ideal and describe their parallel implementation in SINGULAR.
Our modular algorithms to solve problems over Q mainly consist of three parts,
solving the problem modulo p for several primes p, lifting the result to Q by
applying Chinese remainder resp. rational reconstruction, and a part of
verification. Arnold proved using the Hilbert function that the verification
part in the modular algorithm to compute Gr\"obner bases can be simplified for
homogeneous ideals (cf. \cite{A03}). The idea of the proof could easily be
adapted to the local case, i.e. for local orderings and not necessarily
homogeneous ideals, using the Hilbert-Samuel function (cf. \cite{Pf07}). In
this paper we prove the corresponding theorem for non-homogeneous ideals in
case of a global ordering.Comment: 16 page
Parallel algorithms for normalization
Given a reduced affine algebra A over a perfect field K, we present parallel
algorithms to compute the normalization \bar{A} of A. Our starting point is the
algorithm of Greuel, Laplagne, and Seelisch, which is an improvement of de
Jong's algorithm. First, we propose to stratify the singular locus Sing(A) in a
way which is compatible with normalization, apply a local version of the
normalization algorithm at each stratum, and find \bar{A} by putting the local
results together. Second, in the case where K = Q is the field of rationals, we
propose modular versions of the global and local-to-global algorithms. We have
implemented our algorithms in the computer algebra system SINGULAR and compare
their performance with that of the algorithm of Greuel, Laplagne, and Seelisch.
In the case where K = Q, we also discuss the use of modular computations of
Groebner bases, radicals, and primary decompositions. We point out that in most
examples, the new algorithms outperform the algorithm of Greuel, Laplagne, and
Seelisch by far, even if we do not run them in parallel.Comment: 19 page
Improved algorithm for computing separating linear forms for bivariate systems
We address the problem of computing a linear separating form of a system of
two bivariate polynomials with integer coefficients, that is a linear
combination of the variables that takes different values when evaluated at the
distinct solutions of the system. The computation of such linear forms is at
the core of most algorithms that solve algebraic systems by computing rational
parameterizations of the solutions and this is the bottleneck of these
algorithms in terms of worst-case bit complexity. We present for this problem a
new algorithm of worst-case bit complexity \sOB(d^7+d^6\tau) where and
denote respectively the maximum degree and bitsize of the input (and
where \sO refers to the complexity where polylogarithmic factors are omitted
and refers to the bit complexity). This algorithm simplifies and
decreases by a factor the worst-case bit complexity presented for this
problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields,
for this problem, a probabilistic Las-Vegas algorithm of expected bit
complexity \sOB(d^5+d^4\tau).Comment: ISSAC - 39th International Symposium on Symbolic and Algebraic
Computation (2014
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