6,632 research outputs found
On the computation of rational points of a hypersurface over a finite field
We design and analyze an algorithm for computing rational points of
hypersurfaces defined over a finite field based on searches on "vertical
strips", namely searches on parallel lines in a given direction. Our results
show that, on average, less than two searches suffice to obtain a rational
point. We also analyze the probability distribution of outputs, using the
notion of Shannon entropy, and prove that the algorithm is somewhat close to
any "ideal" equidistributed algorithm.Comment: 31 pages, 5 table
The probability that a complete intersection is smooth
Given a smooth subscheme of a projective space over a finite field, we
compute the probability that its intersection with a fixed number of
hypersurface sections of large degree is smooth of the expected dimension. This
generalizes the case of a single hypersurface, due to Poonen. We use this
result to give a probabilistic model for the number of rational points of such
a complete intersection. A somewhat surprising corollary is that the number of
rational points on a random smooth intersection of two surfaces in projective
3-space is strictly less than the number of points on the projective line.Comment: 14 pages; v3: final journal versio
Rational points on certain hyperelliptic curves over finite fields
Let be a field, and . Let us consider the
polynomials , where is a fixed
positive integer. In this paper we show that for each the
hypersurface given by the equation \begin{equation*} S_{k}^{i}:
u^2=\prod_{j=1}^{k}g_{i}(x_{j}),\quad i=1, 2. \end{equation*} contains a
rational curve. Using the above and Woestijne's recent results \cite{Woe} we
show how one can construct a rational point different from the point at
infinity on the curves defined over a finite
field, in polynomial time.Comment: Revised version will appear in Bull. Polish Acad. Sci. Mat
An Algorithm to Compute the Topological Euler Characteristic, Chern-Schwartz-MacPherson Class and Segre Class of Projective Varieties
Let be a closed subscheme of a projective space . We give
an algorithm to compute the Chern-Schwartz-MacPherson class, Euler
characteristic and Segre class of . The algorithm can be implemented using
either symbolic or numerical methods. The algorithm is based on a new method
for calculating the projective degrees of a rational map defined by a
homogeneous ideal. Using this result and known formulas for the
Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre
class of a projective variety in terms of the projective degrees of certain
rational maps we give algorithms to compute the Chern-Schwartz-MacPherson class
and Segre class of a projective variety. Since the Euler characteristic of
is the degree of the zero dimensional component of the
Chern-Schwartz-MacPherson class of our algorithm also computes the Euler
characteristic . Relationships between the algorithm developed here
and other existing algorithms are discussed. The algorithm is tested on several
examples and performs favourably compared to current algorithms for computing
Chern-Schwartz-MacPherson classes, Segre classes and Euler characteristics
Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements
We investigate the interplay between the monodromy and the Deligne mixed
Hodge structure on the Milnor fiber of a homogeneous polynomial. In the case of
hyperplane arrangement Milnor fibers, we obtain a new result on the possible
weights. For line arrangements, we prove in a new way the fact due to Budur and
Saito that the spectrum is determined by the weak combinatorial data, and show
that such a result fails for the Hodge-Deligne polynomials.Comment: An appendix is added in this second version, where we use -adic
Hodge theory to prove that quite generally, whenever a \G-variety is
defined over a number field, the number of rational points of its reductions
modulo prime ideals can be used in certain cases to compute the equivariant
Hodge-Deligne polynomia
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