3,272 research outputs found
Direction problems in affine spaces
This paper is a survey paper on old and recent results on direction problems
in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October
5, 2012, Brussels, Belgiu
Configurations of lines and models of Lie algebras
The automorphism groups of the 27 lines on the smooth cubic surface or the 28
bitangents to the general quartic plane curve are well-known to be closely
related to the Weyl groups of and . We show how classical
subconfigurations of lines, such as double-sixes, triple systems or Steiner
sets, are easily constructed from certain models of the exceptional Lie
algebras. For and we are lead to
beautiful models graded over the octonions, which display these algebras as
plane projective geometries of subalgebras. We also interpret the group of the
bitangents as a group of transformations of the triangles in the Fano plane,
and show how this allows to realize the isomorphism in terms of harmonic cubes.Comment: 31 page
(2^n,2^n,2^n,1)-relative difference sets and their representations
We show that every -relative difference set in
relative to can be represented by a polynomial f(x)\in \F_{2^n}[x],
where is a permutation for each nonzero . We call such an
a planar function on \F_{2^n}. The projective plane obtained from
in the way of Ganley and Spence \cite{ganley_relative_1975} is
coordinatized, and we obtain necessary and sufficient conditions of to be
a presemifield plane. We also prove that a function on \F_{2^n} with
exactly two elements in its image set and is planar, if and only if,
for any x,y\in\F_{2^n}
How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties
Let denote the discriminant variety of degree
polynomials in one variable with at least one of its roots being of
multiplicity . We prove that the tangent cones to
span thus, revealing an extreme ruled nature of these
varieties. The combinatorics of the web of affine tangent spaces to in is directly linked to the root multiplicities
of the relevant polynomials. In fact, solving a polynomial equation
turns out to be equivalent to finding hyperplanes through a given point
P(z)\in \mathcal D_{d,1} \approx \A^d which are tangent to the discriminant
hypersurface . We also connect the geometry of the Vi\`{e}te
map \mathcal V_d: \A^d_{root} \to \A^d_{coef}, given by the elementary
symmetric polynomials, with the tangents to the discriminant varieties
.
Various -partitions provide a refinement of the stratification of \A^d_{coef} by the 's. Our main result, Theorem 7.1, describes an intricate relation
between the divisibility of polynomials in one variable and the families of
spaces tangent to various strata .Comment: 43 pages, 12 figure
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
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