Let k be a field of characteristic not equal to two, B a smooth projective curve of genus g(B) over k, and F its function field. A quadric hypersurface fibration is a flat projective morphism π : X → B such that each geometric fiber is a quadric hypersurface with at worst an isolated singularity and the generic fiber is smooth. Sections σ : B → X of π are in bijection with rational points X(F )...

Let C be a smooth projective irreducible curve defined over a finite field k. Let G be a finite subgroup of Aut(C) of order prime to char(k) and M the |G|-th cyclotomic field. We show that certain idempotent relations in M [G] imply relations among arithmetic invariants of the fundamental groups of the quotient curves (C/H)×k k, where H is a subgroup of G.

Let Fp = Z/pZ. The height of a point a = (a1, . . . , ad) ∈ F d p is hp(a) = min n Pd i=1(kai mod p) : k = 1, . . . , p − 1 o . Explicit formulas and estimates are obtained for the values of the height function in the case d = 2, and these results are applied to the problem of determining the minimum number of edges the must be deleted from a finite directed graph so that the resulting subgraph...

There exists a function f : N → N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension ≥ f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois gro...

Let X c pn be an irreducible projective variety defined over the finite field F q. We will say that X satisfies the property F F N (k, q) (Finite Field Nullstellensatz of degree k over F q) if every homogeneous polynomial over F q of degree k on pn vanishing on X (F q) vanishes on X, that is, it vanishes at all points of X defined over the algebraic closure of F q . We study when FFN(k,q) holds...

In the early eighties tools from algebraic geometry were applied by V. Goppa to construct linear codes using algebraic curves over finite fields, see [7]. Nowadays these codes are called algebraic-geometric codes, AG codes for short. The starting point in the construction of an AG code is a projective, absolutely irreducible, non singular algebraic curve X of genus g ≥ 1 defined over the finite...

We construct canonical heights of subvarieties for dynamical systems of several morphisms associated with line bundles defined over a number field, and study some of their properties. We also construct invariant currents for such systems over C. Introduction Let X be a projective variety over a field K and fi : X → X (i = 1, · · · , k) morphisms over K. Let L be a line bundle on X, and d > k a ...

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 K that induce planar functions on infinitely many extensions of K; we call such polynomials exceptional planar. Exceptional planar monomials have been recently classified. In this paper we establish a partial...

There exists a function f : N → N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension ≥ f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois gro...

New constructions of 3-dimensional optical orthogonal codes are presented. In each case the codes have ideal autocorrelation λa = 0, and cross correlation of λc = 1. All codes produced are demonstrated to be optimal. The constructions utilize a particular automorphism (a Singer cycle) of PG(k,q), the finite projective geometry of dimension k over the field of order q, or its affine analogue in ...