Improved dense multivariate polynomial factorization algorithms
نویسنده
چکیده
We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in sub-quadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the number of variables is at least three. We also investigate the reduction from several to two variables and improve the quantitative version of Bertini’s irreducibility theorem.
منابع مشابه
Lifting and recombination techniques for absolute factorization
In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.
متن کاملChallenges in Polynomial Factorization∗
Algebraic complexity theory studies the complexity of computing (multivariate) polynomials efficiently using algebraic circuits. This succinct representation leads to fundamental algorithmic challenges such as the polynomial identity testing (PIT) problem (decide non-zeroness of the computed polynomial) and the polynomial factorization problem (compute succinct representations of the factors of...
متن کاملParallel Polynomial Operations on SMPs: an Overview
1 SMP-based parallel algorithms and implementations for polynomial factoring and GCD are overviewed. Topics include polynomial factoring modulo small primes, univariate and multivariate p-adic lifting, and reformulation of lift basis. Sparse polynomial GCD is also covered.
متن کاملFast polynomial factorization, modular composition, and multipoint evaluation of multivariate polynomials in small characteristic
We obtain randomized algorithms for factoring degree n univariate polynomials over Fq that use O(n + n log q) field operations, when the characteristic is at most n. When log q < n, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for log q ≥ n, it matches the asymptotic running time of the best known algorithms. The im...
متن کاملTwo Families of Algorithms for Symbolic Polynomials
We consider multivariate polynomials with exponents that are themselves integer-valued multivariate polynomials, and we present algorithms to compute their GCD and factorization. The algorithms fall into two families: algebraic extension methods and projection methods. The first family of algorithms uses the algebraic independence of x, x, x 2 , x, etc, to solve related problems with more indet...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- J. Symb. Comput.
دوره 42 شماره
صفحات -
تاریخ انتشار 2007