5,593 research outputs found
Factoring Polynomials over Finite Fields using Balance Test
We study the problem of factoring univariate polynomials over finite fields.
Under the assumption of the Extended Riemann Hypothesis (ERH), (Gao, 2001)
designed a polynomial time algorithm that fails to factor only if the input
polynomial satisfies a strong symmetry property, namely square balance. In this
paper, we propose an extension of Gao's algorithm that fails only under an even
stronger symmetry property. We also show that our property can be used to
improve the time complexity of best deterministic algorithms on most input
polynomials. The property also yields a new randomized polynomial time
algorithm
Character Sums and Deterministic Polynomial Root Finding in Finite Fields
We obtain a new bound of certain double multiplicative character sums. We use
this bound together with some other previously obtained results to obtain new
algorithms for finding roots of polynomials modulo a prime
Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra
We present algorithms to factorize weighted homogeneous elements in the first
polynomial Weyl algebra and -Weyl algebra, which are both viewed as a
-graded rings. We show, that factorization of homogeneous
polynomials can be almost completely reduced to commutative univariate
factorization over the same base field with some additional uncomplicated
combinatorial steps. This allows to deduce the complexity of our algorithms in
detail. Furthermore, we will show for homogeneous polynomials that
irreducibility in the polynomial first Weyl algebra also implies irreducibility
in the rational one, which is of interest for practical reasons. We report on
our implementation in the computer algebra system \textsc{Singular}. It
outperforms for homogeneous polynomials currently available implementations
dealing with factorization in the first Weyl algebra both in speed and elegancy
of the results.Comment: 26 pages, Singular implementation, 2 algorithms, 1 figure, 2 table
Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree
In this paper we study the problem of deterministic factorization of sparse
polynomials. We show that if is a
polynomial with monomials, with individual degrees of its variables bounded
by , then can be deterministically factored in time . Prior to our work, the only efficient factoring algorithms known for
this class of polynomials were randomized, and other than for the cases of
and , only exponential time deterministic factoring algorithms were
known.
A crucial ingredient in our proof is a quasi-polynomial sparsity bound for
factors of sparse polynomials of bounded individual degree. In particular we
show if is an -sparse polynomial in variables, with individual
degrees of its variables bounded by , then the sparsity of each factor of
is bounded by . This is the first nontrivial bound on
factor sparsity for . Our sparsity bound uses techniques from convex
geometry, such as the theory of Newton polytopes and an approximate version of
the classical Carath\'eodory's Theorem.
Our work addresses and partially answers a question of von zur Gathen and
Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the
sparsity of factors of sparse polynomials
The Minimal Resultant Locus
Let K be a complete, algebraically closed nonarchimedean valued field, and
let f(z) in K(z) be a rational function of degree d at least 2. We give an
algorithm to determine whether f(z) has potential good reduction over K, based
on a geometric reformulation of the problem using the Berkovich Projective
Line. We show the minimal resultant is is either achieved at a single point in
the Berkovich line, or on a segment, and that minimal resultant locus is
contained in the tree in spanned by the fixed points and the poles of f(z).
When f(z) is defined over the rationals, the algorithm runs in probabilistic
polynomial time. If f(z) has potential good reduction, and is defined over a
subfield H of K, we show there is an extension L/H in K with degree at most (d
+ 1)^2 such that f(z) achieves good reduction over L.Comment: 37 page
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