Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the ... more Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the p-adic integers). Current implementations Hensel lift to determine f1,...,fr with a p-adic accuracy a that is guaranteed to be high enough to recover any potential factor of f in Z[x]. However, the problem is that this p-adic accuracy, a, is often much higher than what was actually necessary to recover all the factors g1,...,g k. This implies that current implementations often waste CPU time on Hensel Lifting. In practice it frequently happens that f has one large factor, say g1, and zero or more small factors, say g2,...,g k. Then, to recover g1,...,g k we do not need p a to be larger than twice the largest coefficient of g1. All we need is that p a is larger than twice the largest coefficient in g2,...,g k. This suffices to reconstruct g2,...,g k ∈ Z[x] from their modular images, after which the remaining factor g1 can be determined by a division in Z[x]. It is easy to give examples wh...
Proceedings of the 1996 international symposium on Symbolic and algebraic computation - ISSAC '96, 1996
The topic of this paper is a fast method to compute the rational solutions of a certain different... more The topic of this paper is a fast method to compute the rational solutions of a certain differential equation that will be called the mixed differential equation. This can be applied to speed up the factorization of completely reducible linear differential operators with rational functions coefficients.
Differential equations with 2F1-type solutions are very common in Mathematics and they also occur... more Differential equations with 2F1-type solutions are very common in Mathematics and they also occur in several areas such as Physics and Combinatorics. We are interested in finding 2F1-type solutions of second order differential equations. Paper [1] gives an algorithm to find 2F1-type solutions of second order differential equations when the degree of the pullback function1 of f is three. In our algorithm, the pullback function f is of arbitrary degree (however, we restrict to a special case defined below).
Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94, 1994
In this paper I want to present a new method for computing parametrizations of algebraic curves. ... more In this paper I want to present a new method for computing parametrizations of algebraic curves. Basically this method is a direct application of integral basis computation. Examples show that this method is faster than older methods.
The software is an implementation of the algorithms in [1], [2], and [3]. The main algorithm from... more The software is an implementation of the algorithms in [1], [2], and [3]. The main algorithm from [3] is implemented with additional base equations beyond what appear in [3] and is incorporated into [4]. Common to each algorithm is a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gt-transformations).
Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the ... more Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the p-adic integers). Current implementations Hensel lift to determine f1,...,fr with a p-adic accuracy a that is guaranteed to be high enough to recover any potential factor of f in Z[x]. However, the problem is that this p-adic accuracy, a, is often much higher than what was actually necessary to recover all the factors g1,...,g k. This implies that current implementations often waste CPU time on Hensel Lifting. In practice it frequently happens that f has one large factor, say g1, and zero or more small factors, say g2,...,g k. Then, to recover g1,...,g k we do not need p a to be larger than twice the largest coefficient of g1. All we need is that p a is larger than twice the largest coefficient in g2,...,g k. This suffices to reconstruct g2,...,g k ∈ Z[x] from their modular images, after which the remaining factor g1 can be determined by a division in Z[x]. It is easy to give examples wh...
Proceedings of the 1996 international symposium on Symbolic and algebraic computation - ISSAC '96, 1996
The topic of this paper is a fast method to compute the rational solutions of a certain different... more The topic of this paper is a fast method to compute the rational solutions of a certain differential equation that will be called the mixed differential equation. This can be applied to speed up the factorization of completely reducible linear differential operators with rational functions coefficients.
Differential equations with 2F1-type solutions are very common in Mathematics and they also occur... more Differential equations with 2F1-type solutions are very common in Mathematics and they also occur in several areas such as Physics and Combinatorics. We are interested in finding 2F1-type solutions of second order differential equations. Paper [1] gives an algorithm to find 2F1-type solutions of second order differential equations when the degree of the pullback function1 of f is three. In our algorithm, the pullback function f is of arbitrary degree (however, we restrict to a special case defined below).
Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94, 1994
In this paper I want to present a new method for computing parametrizations of algebraic curves. ... more In this paper I want to present a new method for computing parametrizations of algebraic curves. Basically this method is a direct application of integral basis computation. Examples show that this method is faster than older methods.
The software is an implementation of the algorithms in [1], [2], and [3]. The main algorithm from... more The software is an implementation of the algorithms in [1], [2], and [3]. The main algorithm from [3] is implemented with additional base equations beyond what appear in [3] and is incorporated into [4]. Common to each algorithm is a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gt-transformations).
Uploads
Papers by Mark Hoeij