181 research outputs found
Factoring bivariate sparse (lacunary) polynomials
We present a deterministic algorithm for computing all irreducible factors of
degree of a given bivariate polynomial over an algebraic
number field and their multiplicities, whose running time is polynomial in
the bit length of the sparse encoding of the input and in . Moreover, we
show that the factors over \Qbarra of degree which are not binomials
can also be computed in time polynomial in the sparse length of the input and
in .Comment: 20 pp, Latex 2e. We learned on January 23th, 2006, that a
multivariate version of Theorem 1 had independently been achieved by Erich
Kaltofen and Pascal Koira
On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
We present an efficient quantum algorithm for the exact evaluation of either
the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function
Z for a family of graphs related to irreducible cyclic codes. This problem is
related to the evaluation of the Jones and Tutte polynomials. We consider the
connection between the weight enumerator polynomial from coding theory and Z
and exploit the fact that there exists a quantum algorithm for efficiently
estimating Gauss sums in order to obtain the weight enumerator for a certain
class of linear codes. In this way we demonstrate that for a certain class of
sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_\epsilon)
graphs, quantum computers provide a polynomial speed up in the difference
between the number of edges and vertices of the graph, and an exponential speed
up in q, over the best classical algorithms known to date
Cyclotomic Identity Testing and Applications
We consider the cyclotomic identity testing problem: given a polynomial
, decide whether is
zero, for a primitive complex -th root of unity and
integers . We assume that and are
represented in binary and consider several versions of the problem, according
to the representation of . For the case that is given by an algebraic
circuit we give a randomized polynomial-time algorithm with two-sided errors,
showing that the problem lies in BPP. In case is given by a circuit of
polynomially bounded syntactic degree, we give a randomized algorithm with
two-sided errors that runs in poly-logarithmic parallel time, showing that the
problem lies in BPNC. In case is given by a depth-2 circuit
(or, equivalently, as a list of monomials), we show that the cyclotomic
identity testing problem lies in NC. Under the generalised Riemann hypothesis,
we are able to extend this approach to obtain a polynomial-time algorithm also
for a very simple subclass of depth-3 circuits. We complement
this last result by showing that for a more general class of depth-3
circuits, a polynomial-time algorithm for the cyclotomic
identity testing problem would yield a sub-exponential-time algorithm for
polynomial identity testing. Finally, we use cyclotomic identity testing to
give a new proof that equality of compressed strings, i.e., strings presented
using context-free grammars, can be decided in coRNC: randomized NC with
one-sided errors
Efficiently Detecting Torsion Points and Subtori
Suppose X is the complex zero set of a finite collection of polynomials in
Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose
coordinates are d_th roots of unity can be done within NP^NP (relative to the
sparse encoding), under a plausible assumption on primes in arithmetic
progression. In particular, our hypothesis can still hold even under certain
failures of the Generalized Riemann Hypothesis, such as the presence of
Siegel-Landau zeroes. Furthermore, we give a similar (but UNconditional)
complexity upper bound for n=1. Finally, letting T be any algebraic subgroup of
(C^*)^n we show that deciding whether X contains T is coNP-complete (relative
to an even more efficient encoding),unconditionally. We thus obtain new
non-trivial families of multivariate polynomial systems where deciding the
existence of complex roots can be done unconditionally in the polynomial
hierarchy -- a family of complexity classes lying between PSPACE and P,
intimately connected with the P=?NP Problem. We also discuss a connection to
Laurent's solution of Chabauty's Conjecture from arithmetic geometry.Comment: 21 pages, no figures. Final version, with additional commentary and
references. Also fixes a gap in Theorems 2 (now Theorem 1.3) regarding
translated subtor
New Bounds on Quotient Polynomials with Applications to Exact Divisibility and Divisibility Testing of Sparse Polynomials
A sparse polynomial (also called a lacunary polynomial) is a polynomial that
has relatively few terms compared to its degree. The sparse-representation of a
polynomial represents the polynomial as a list of its non-zero terms
(coefficient-degree pairs). In particular, the degree of a sparse polynomial
can be exponential in the sparse-representation size.
We prove that for monic polynomials such that
divides , the -norm of the quotient polynomial is bounded by
. This improves upon the exponential (in
) bounds for general polynomials and implies that the trivial
long division algorithm runs in time quasi-linear in the input size and number
of terms of the quotient polynomial , thus solving a long-standing problem
on exact divisibility of sparse polynomials.
We also study the problem of bounding the number of terms of in some
special cases. When and is a cyclotomic-free
(i.e., it has no cyclotomic factors) trinomial, we prove that
. When is a binomial with , we
prove that the sparsity is at most . Both upper bounds
are polynomial in the input-size. We leverage these results and give a
polynomial time algorithm for deciding whether a cyclotomic-free trinomial
divides a sparse polynomial over the integers.
As our last result, we present a polynomial time algorithm for testing
divisibility by pentanomials over small finite fields when
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