9,939 research outputs found
On some approximation problems concerning sparse polynomials over finite fields
AbstractWe obtain new lower bounds on the number of non-zeros of sparse polynomials and give a fully polynomial time (ε, δ) approximation algorithm for the number of non-zeros of multivariate sparse polynomials over a finite field of q elements and degree less than q − 1. This partially answers an open problem of D. Grigoriev and M. Karpinski. Also, probabilistic and deterministic algorithms for testing identity to zero of a sparse polynomial given by a “black-box” are given. Finally, we propose an algorithm to estimate the size of the image of a univariate sparse polynomial
Elimination for generic sparse polynomial systems
We present a new probabilistic symbolic algorithm that, given a variety
defined in an n-dimensional affine space by a generic sparse system with fixed
supports, computes the Zariski closure of its projection to an l-dimensional
coordinate affine space with l < n. The complexity of the algorithm depends
polynomially on combinatorial invariants associated to the supports.Comment: 22 page
Computing sparse multiples of polynomials
We consider the problem of finding a sparse multiple of a polynomial. Given f
in F[x] of degree d over a field F, and a desired sparsity t, our goal is to
determine if there exists a multiple h in F[x] of f such that h has at most t
non-zero terms, and if so, to find such an h. When F=Q and t is constant, we
give a polynomial-time algorithm in d and the size of coefficients in h. When F
is a finite field, we show that the problem is at least as hard as determining
the multiplicative order of elements in an extension field of F (a problem
thought to have complexity similar to that of factoring integers), and this
lower bound is tight when t=2.Comment: Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 650
Efficient Computation of the Characteristic Polynomial
This article deals with the computation of the characteristic polynomial of
dense matrices over small finite fields and over the integers. We first present
two algorithms for the finite fields: one is based on Krylov iterates and
Gaussian elimination. We compare it to an improvement of the second algorithm
of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third
algorithm could improve both complexity and computational time. We use these
results as a basis for the computation of the characteristic polynomial of
integer matrices. We first use early termination and Chinese remaindering for
dense matrices. Then a probabilistic approach, based on integer minimal
polynomial and Hensel factorization, is particularly well suited to sparse
and/or structured matrices
A note on Probably Certifiably Correct algorithms
Many optimization problems of interest are known to be intractable, and while
there are often heuristics that are known to work on typical instances, it is
usually not easy to determine a posteriori whether the optimal solution was
found. In this short note, we discuss algorithms that not only solve the
problem on typical instances, but also provide a posteriori certificates of
optimality, probably certifiably correct (PCC) algorithms. As an illustrative
example, we present a fast PCC algorithm for minimum bisection under the
stochastic block model and briefly discuss other examples
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