125 research outputs found
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
A Comparison of Two Families of Algorithms for Symbolic Polynomials
Symbolic polynomials, whose exponents themselves are integer-valued multivariate polynomials, arise often in algorithm analysis. Unfortunately, modern computer algebra systems do not provide ample support for their algebraic structures. Basic operations involving symbolic polynomials are indeed trivial (addition, multiplication, derivatives); however, other crucial operations remain much more difficult, such as factorization and GCD. Watt has given two separate methods to solve these challenging problems. The first uses a change of variables, e.g. xn to X, xnm to Y. Doing so increases the number of variables potentially exponentially. Evaluation/Interpolation is used in the second algorithm. Projection methods introduce fewer variables but have larger degrees. We propose several adaptations to this idea to attempt to reduce the degree and number of required images. These include: sparse interpolation, evaluation point optimization and evaluating at primitive roots of unity. We give an in depth comparison of these algorithms, both empirically and theoretically
Development of symbolic algorithms for certain algebraic processes
This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested
Resultants and subresultants of p-adic polynomials
We address the problem of the stability of the computations of resultants and
subresultants of polynomials defined over complete discrete valuation rings
(e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms
are highly unstable on average and we explain, in many cases, how one can
stabilize them without sacrifying the complexity. On the way, we completely
determine the distribution of the valuation of the principal subresultants of
two random monic p-adic polynomials having the same degree
Fast Arithmetics Using Chinese Remaindering
In this paper, some issues concerning the Chinese remaindering representation
are discussed. Some new converting methods, including an efficient
probabilistic algorithm based on a recent result of von zur Gathen and
Shparlinski \cite{Gathen-Shparlinski}, are described. An efficient refinement
of the NC division algorithm of Chiu, Davida and Litow
\cite{Chiu-Davida-Litow} is given, where the number of moduli is reduced by a
factor of
A deterministic version of Pollard's p-1 algorithm
In this article we present applications of smooth numbers to the
unconditional derandomization of some well-known integer factoring algorithms.
We begin with Pollard's algorithm, which finds in random polynomial
time the prime divisors of an integer such that is smooth. We
show that these prime factors can be recovered in deterministic polynomial
time. We further generalize this result to give a partial derandomization of
the -th cyclotomic method of factoring () devised by Bach and
Shallit.
We also investigate reductions of factoring to computing Euler's totient
function . We point out some explicit sets of integers that are
completely factorable in deterministic polynomial time given . These
sets consist, roughly speaking, of products of primes satisfying, with the
exception of at most two, certain conditions somewhat weaker than the
smoothness of . Finally, we prove that oracle queries for
values of are sufficient to completely factor any integer in less
than deterministic
time.Comment: Expanded and heavily revised version, to appear in Mathematics of
Computation, 21 page
Resolving zero-divisors using Hensel lifting
Algorithms which compute modulo triangular sets must respect the presence of
zero-divisors. We present Hensel lifting as a tool for dealing with them. We
give an application: a modular algorithm for computing GCDs of univariate
polynomials with coefficients modulo a radical triangular set over the
rationals. Our modular algorithm naturally generalizes previous work from
algebraic number theory. We have implemented our algorithm using Maple's RECDEN
package. We compare our implementation with the procedure RegularGcd in the
RegularChains package.Comment: Shorter version to appear in Proceedings of SYNASC 201
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