993 research outputs found
On homogeneous planar functions
Let be an odd prime and \F_q be the finite field with elements.
A planar function f:\F_q\rightarrow\F_q is called homogenous if for all \lambda\in\F_p and x\in\F_q, where is some
fixed positive integer. We characterize as the unique homogenous planar
function over \F_{p^2} up to equivalence.Comment: Introduction modified to: 1. give the correct definition of
equivalence, 2. add some references. Other part unaltere
Exceptional planar polynomials
Planar functions are special functions from a finite field to itself that
give rise to finite projective planes and other combinatorial objects. We
consider polynomials over a finite field that induce planar functions on
infinitely many extensions of ; we call such polynomials exceptional planar.
Exceptional planar monomials have been recently classified. In this paper we
establish a partial classification of exceptional planar polynomials. This
includes results for the classical planar functions on finite fields of odd
characteristic and for the recently proposed planar functions on finite fields
of characteristic two
A new family of semifields with 2 parameters
A new family of commutative semifields with two parameters is presented. Its
left and middle nucleus are both determined. Furthermore, we prove that for any
different pairs of parameters, these semifields are not isotopic. It is also
shown that, for some special parameters, one semifield in this family can lead
to two inequivalent planar functions. Finally, using similar construction, new
APN functions are given
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
Irreducibility properties of Keller maps
Jedrzejewicz showed that a polynomial map over a field of characteristic zero
is invertible, if and only if the corresponding endomorphism maps irreducible
polynomials to irreducible polynomials. Furthermore, he showed that a
polynomial map over a field of characteristic zero is a Keller map, if and only
if the corresponding endomorphism maps irreducible polynomials to square-free
polynomials. We show that the latter endomorphism maps other square-free
polynomials to square-free polynomials as well.
In connection with the above classification of invertible polynomial maps and
the Jacobian Conjecture, we study irreducible properties of several types of
Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith,
we generalize the result of Bakalarski, that the components of cubic
homogeneous Keller maps with a symmetric Jacobian matrix (over C and hence any
field of characteristic zero) are irreducible.
Furthermore, we show that the Jacobian Conjecture can even be reduced to any
of these types with the extra condition that each affinely linear combination
of the components of the polynomial map is irreducible. This is somewhat
similar to reducing the planar Jacobian Conjecture to the so-called (planar)
weak Jacobian Conjecture by Kaliman.Comment: 22 page
Discrete phase-space approach to mutually orthogonal Latin squares
We show there is a natural connection between Latin squares and commutative
sets of monomials defining geometric structures in finite phase-space of prime
power dimensions. A complete set of such monomials defines a mutually unbiased
basis (MUB) and may be associated with a complete set of mutually orthogonal
Latin squares (MOLS). We translate some possible operations on the monomial
sets into isomorphisms of Latin squares, and find a general form of
permutations that map between Latin squares corresponding to unitarily
equivalent mutually unbiased sets. We extend this result to a conjecture: MOLS
associated to unitarily equivalent MUBs will always be isomorphic, and MOLS
associated to unitarily inequivalent MUBs will be non-isomorphic
- …