5,818 research outputs found
Sparse Rational Univariate Representation
International audienceWe present explicit worst case degree and height bounds for the rational univariate representation of the isolated roots of polynomial systems based on mixed volume. We base our estimations on height bounds of resultants and we consider the case of 0-dimensional, positive dimensional, and parametric polynomial systems
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
Reconstructing Rational Functions with
We present the open-source library for the
reconstruction of multivariate rational functions over finite fields. We
discuss the involved algorithms and their implementation. As an application, we
use in the context of integration-by-parts reductions and
compare runtime and memory consumption to a fully algebraic approach with the
program .Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio
Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants
We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.
Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring).
All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface.
For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system.
This yields an efficient, output-sensitive algorithm for
computing the discriminant polynomial
Counting Value Sets: Algorithm and Complexity
Let be a prime. Given a polynomial in \F_{p^m}[x] of degree over
the finite field \F_{p^m}, one can view it as a map from \F_{p^m} to
\F_{p^m}, and examine the image of this map, also known as the value set. In
this paper, we present the first non-trivial algorithm and the first complexity
result on computing the cardinality of this value set. We show an elementary
connection between this cardinality and the number of points on a family of
varieties in affine space. We then apply Lauder and Wan's -adic
point-counting algorithm to count these points, resulting in a non-trivial
algorithm for calculating the cardinality of the value set. The running time of
our algorithm is . In particular, this is a polynomial time
algorithm for fixed if is reasonably small. We also show that the
problem is #P-hard when the polynomial is given in a sparse representation,
, and is allowed to vary, or when the polynomial is given as a
straight-line program, and is allowed to vary. Additionally, we prove
that it is NP-hard to decide whether a polynomial represented by a
straight-line program has a root in a prime-order finite field, thus resolving
an open problem proposed by Kaltofen and Koiran in
\cite{Kaltofen03,KaltofenKo05}
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