Journal Article10.1137/0213050
Parallel algorithms for algebraic problems
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TL;DR: Fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials: computing all entries of the extended Euclidean scheme of two polynmials over an arbitrary field, gcd and 1cm of many poynomials, factoring polynomsials over finite fields, and the squarefree decomposition of polynOMials over fields of characteristic zero and over infinite fields.
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Abstract: Fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials: computing all entries of the extended Euclidean scheme of two polynomials over an arbitrary field, gcd and 1cm of many polynomials, factoring polynomials over finite fields, and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields.For the following estimates, assume that the input polynomials have degree at most n, and the finite field has $p^d $ elements. The Euclidean algorithm is deterministic and runs in parallel time $O(\log ^2 n)$. All the other algorithms are probabilistic (Las Vegas) in the general case, but when applicable to ${\bf Q}$ or ${\bf R}$, they can be implemented deterministically over these fields. The algorithms for gcd and lcm use parallel time $O(\log ^2 n)$. The factoring algorithm runs in parallel time $O(\log ^2 n\log ^2 (d + 1)\log p)$. The algorithm for squarefree decomposition runs in parallel time $O(\log ^2 n)$...
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Computing with polynomials given byblack boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators
Erich Kaltofen,Barry M. Trager +1 more
TL;DR: It is shown that within this black box representation the polynomial greatest common divisor and factorization problems, as well as the problem of extracting the numerator and denominator of a rational function, can all be solved in randomPolynomial-time.
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Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators
Erich Kaltofen,B. Trager +1 more
- 24 Oct 1988
TL;DR: In this article, a black-box representation for multivariate polynomials and rational functions with rational coefficients was developed for their evaluation, and it was shown that within this representation, the polynomial greatest common divisor and factorization problems as well as the problem of extracting the numerator and denominator of a rational function can be solved in random time in the usual parameters.
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