1,540 research outputs found
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
On the complexity of solving linear congruences and computing nullspaces modulo a constant
We consider the problems of determining the feasibility of a linear
congruence, producing a solution to a linear congruence, and finding a spanning
set for the nullspace of an integer matrix, where each problem is considered
modulo an arbitrary constant k>1. These problems are known to be complete for
the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case
that k is prime (Buntrock et al, 1992). By considering variants of standard
logspace function classes --- related to #L and functions computable by UL
machines, but which only characterize the number of accepting paths modulo k
--- we show that these problems of linear algebra are also complete for
{coMod_k L} for any constant k>1.
Our results are obtained by defining a class of functions FUL_k which are low
for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the
case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1
reductions (including {Mod_k L} oracle reductions). In addition to the results
above, we briefly consider the relationship of the class FUL_k for arbitrary
moduli k to the class {F.coMod_k L} of functions whose output symbols are
verifiable by {coMod_k L} algorithms; and consider what consequences such a
comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L}
= {Mod_k L} for composite k.Comment: 17 pages, one Appendix; minor corrections and revisions to
presentation, new observations regarding the prospect of oracle closures.
Comments welcom
Integer Division by Constants: Optimal Bounds
The integer division of a numerator n by a divisor d gives a quotient q and a
remainder r. Optimizing compilers accelerate software by replacing the division
of n by d with the division of c * n (or c * n + c) by m for convenient
integers c and m chosen so that they approximate the reciprocal: c/m ~= 1/d.
Such techniques are especially advantageous when m is chosen to be a power of
two and when d is a constant so that c and m can be precomputed. The literature
contains many bounds on the distance between c/m and the divisor d. Some of
these bounds are optimally tight, while others are not. We present optimally
tight bounds for quotient and remainder computations
Faster polynomial multiplication over finite fields
Let p be a prime, and let M_p(n) denote the bit complexity of multiplying two
polynomials in F_p[X] of degree less than n. For n large compared to p, we
establish the bound M_p(n) = O(n log n 8^(log^* n) log p), where log^* is the
iterated logarithm. This is the first known F\"urer-type complexity bound for
F_p[X], and improves on the previously best known bound M_p(n) = O(n log n log
log n log p)
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