8,859 research outputs found
Planar functions over fields of characteristic two
Classical planar functions are functions from a finite field to itself and
give rise to finite projective planes. They exist however only for fields of
odd characteristic. We study their natural counterparts in characteristic two,
which we also call planar functions. They again give rise to finite projective
planes, as recently shown by the second author. We give a characterisation of
planar functions in characteristic two in terms of codes over .
We then specialise to planar monomial functions and present
constructions and partial results towards their classification. In particular,
we show that is the only odd exponent for which is planar
(for some nonzero ) over infinitely many fields. The proof techniques
involve methods from algebraic geometry.Comment: 23 pages, minor corrections and simplifications compared to the first
versio
Comments on the Links between su(3) Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards
We examine the proposal made recently that the su(3) modular invariant
partition functions could be related to the geometry of the complex Fermat
curves. Although a number of coincidences and similarities emerge between them
and certain algebraic curves related to triangular billiards, their meaning
remains obscure. In an attempt to go beyond the su(3) case, we show that any
rational conformal field theory determines canonically a Riemann surface.Comment: 56 pages, 4 eps figures, LaTeX, uses eps
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
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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