Polynomials in Finite Geometry (original) (raw)

We will not be very strict and consistent in the notation (but at least we'll try to be). However, here we give a short description of the typical notation we are going to use. If not specified differently, q = p h is a prime power, p is a prime, and we work in the Desarguesian projective (or affine) space PG(n, q) (AG(n, q), resp.), each space coordinatized by the finite (Galois) field GF(q). The n-dimensional vectorspace over GF(q) will be denoted by V(n, q) or simply by GF(q) n. When discussing PG(n, q) and the related V(n + 1, q) together then for a subspace dimension will be meant rank=dim+1 projectively while vector space dimension will be called rank. A field, which is not necessarily finite will be denoted by F. In general capital letters X, Y, Z, T, ... will denote independent variables, while x, y, z, t, ... will be elements of GF(q). A pair or triple of variables or elements in any pair of brackets can be meant homogeneously, hopefully it will be always clear from the context and the actual setting. We write X or V = (X, Y, Z, ..., T) meaning as many variables as needed; V q = (X q , Y q , Z q , ...). As over a finite field of order q for each x ∈ GF(q) x q = x holds, two different polynomials, f and g, in one or more variables, can have coinciding values "everywhere" over GF(q). In this case we ought to write f ≡ g, as for univariate polynomials f (X), g(X) it means that f ≡ g (mod X q − X) in the ring GF(q)[X]. However, as in the literature f ≡ g is used in the sense 'f and g are equal as polynomials', we will use it in the same sense; though simply f = g and f (X) = g(X) may denote the same. Throughout this book we mostly use the usual representation of PG(n, q). This means that the points have homogeneous coordinates (x, y, z, ..., t) where x, y, z, ..., t are elements of GF(q). The hyperplane [a, b, c, ..., d] of the space have equation aX + bY + cZ + ... + dT = 0. When PG(n, q) is considered as AG(n, q) plus the hyperplane at infinity, then we will use the notation H ∞ for that ('ideal') hyperplane. If n = 2 then H ∞ is called the line at infinity ∞. According to the standard terminology, a line meeting a pointset in one point will be called a tangent and a line intersecting it in r points is an r-secant (or a line of length r). This book is about combinatorially defined (point)sets of (mainly projective or affine) finite geometries. They are defined by their intersection numbers with lines (or other subspaces) typically. The most important definitions and basic information are collected in the Glossary of concepts at the end of this book. These are: blocking sets, arcs, nuclei, spreads, sets of even type, etc. Warning. In this book a curve is allowed to have multiple components, so in fact the curves considered here are called cycles in a different terminology.