34 research outputs found
On fully split lacunary polynomials in finite fields
We estimate the number of possible types degree patterns of -lacunary
polynomials of degree which split completely modulo . The result is
based on a combination of a bound on the number of zeros of lacunary
polynomials with some graph theory arguments.Comment: 8 pages. Bull. Polish Acad. Sci. Math., to appea
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
Bounded-degree factors of lacunary multivariate polynomials
In this paper, we present a new method for computing bounded-degree factors
of lacunary multivariate polynomials. In particular for polynomials over number
fields, we give a new algorithm that takes as input a multivariate polynomial f
in lacunary representation and a degree bound d and computes the irreducible
factors of degree at most d of f in time polynomial in the lacunary size of f
and in d. Our algorithm, which is valid for any field of zero characteristic,
is based on a new gap theorem that enables reducing the problem to several
instances of (a) the univariate case and (b) low-degree multivariate
factorization.
The reduction algorithms we propose are elementary in that they only
manipulate the exponent vectors of the input polynomial. The proof of
correctness and the complexity bounds rely on the Newton polytope of the
polynomial, where the underlying valued field consists of Puiseux series in a
single variable.Comment: 31 pages; Long version of arXiv:1401.4720 with simplified proof
Factoring bivariate lacunary polynomials without heights
We present an algorithm which computes the multilinear factors of bivariate
lacunary polynomials. It is based on a new Gap Theorem which allows to test
whether a polynomial of the form P(X,X+1) is identically zero in time
polynomial in the number of terms of P(X,Y). The algorithm we obtain is more
elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on
the valuation of polynomials of the previous form instead of the height of the
coefficients. As a result, it can be used to find some linear factors of
bivariate lacunary polynomials over a field of large finite characteristic in
probabilistic polynomial time.Comment: 25 pages, 1 appendi
Finite Fields: Theory and Applications
Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation
The Multivariate Resultant is NP-hard in any Characteristic
The multivariate resultant is a fundamental tool of computational algebraic
geometry. It can in particular be used to decide whether a system of n
homogeneous equations in n variables is satisfiable (the resultant is a
polynomial in the system's coefficients which vanishes if and only if the
system is satisfiable). In this paper we present several NP-hardness results
for testing whether a multivariate resultant vanishes, or equivalently for
deciding whether a square system of homogeneous equations is satisfiable. Our
main result is that testing the resultant for zero is NP-hard under
deterministic reductions in any characteristic, for systems of low-degree
polynomials with coefficients in the ground field (rather than in an
extension). We also observe that in characteristic zero, this problem is in the
Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In
positive characteristic, the best upper bound remains PSPACE.Comment: 13 page
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
Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy
We consider the average-case complexity of some otherwise undecidable or open
Diophantine problems. More precisely, consider the following: (I) Given a
polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y
f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given
polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a
rational solution to f_1=...=f_m=0. We show that, for almost all inputs,
problem (I) can be done within coNP. The decidability of problem (I), over N
and Z, was previously unknown. We also show that the Generalized Riemann
Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done
via within the complexity class PP^{NP^NP}, i.e., within the third level of the
polynomial hierarchy. The decidability of problem (II), even in the case m=n=2,
remains open in general.
Along the way, we prove results relating polynomial system solving over C, Q,
and Z/pZ. We also prove a result on Galois groups associated to sparse
polynomial systems which may be of independent interest. A practical
observation is that the aforementioned Diophantine problems should perhaps be
avoided in the construction of crypto-systems.Comment: Slight revision of final journal version of an extended abstract
which appeared in STOC 1999. This version includes significant corrections
and improvements to various asymptotic bounds. Needs cjour.cls to compil
Deterministically Factoring Sparse Polynomials into Multilinear Factors and Sums of Univariate Polynomials
We present the first efficient deterministic algorithm for factoring sparse polynomials that split into multilinear factors
and sums of univariate polynomials. Our result makes partial progress towards the resolution of the classical question posed by von zur Gathen and Kaltofen in [von zur Gathen/Kaltofen, J. Comp. Sys. Sci., 1985] to devise an efficient deterministic algorithm for factoring (general) sparse polynomials. We achieve our goal by introducing essential factorization schemes which can be thought of as a relaxation of the regular factorization notion