Applications of Polynomials Over Finite Fields (original) (raw)

Polynomials in finite geometry

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.

Mathematics Subject Classification. Primary 11G20; Secondary 11T55

2011

In this paper, we discuss in more detail some of the results on the statistics of the trace of the Frobenius endomorphism associated to cyclic p-fold covers of the projective line that were presented in [1]. We also show new findings regarding statistics associated to such curves where we fix the number of zeros in some of the factors of the equation in the affine model.

О производных унимодулярных многочленов

Sbornik: Mathematics, 2016

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let P c n denote the set of all algebraic polynomials of degree at most n with complex coefficients. Associated with λ ≥ 0 let K λ n := p n : p n (z) = n k=0 a k k λ z k , a k ∈ C , |a k | = 1 ⊂ P c n .

Algebraic geometry (Selected Topics)

This topic contains lecture notes from one-semester course for graduate students which took place in Mathematical institute SANU in the spring semester 1994/95. It includes basic notions of algebraic geometry such as plane algebraic curves, Weil and Cartier divisors, dimension, sheaves and Czech cohomology, topological and arithmetical genus, linear systems, Riemann-Roch theorem for curves.

Séminaire de Théorie des Nombres, Paris 1987–88

1990

The method of choice nowadays for achieving a group G as a Galois group of a regular extension of ((x) goes under the heading of rigidity. It works essentially, only, to produce Galois extensions of Q(x) ramified over 3 points. The three rigidity conditions ((0.1) below) imply that G is generated in a very special way by two elements. Generalization of rigidity that considers extensions with any number r of branch points has been around even longer than rigidity { 5.1). Of the three conditions, the generalization of the transitivity condition, 0.1 c), requires only the addition of an action of the Hurwitz monodromy group H (a quotient of the Artin braid group). But it also adds a 4th condition that in many situations amounts to asking for a Q-point on the Hurwitz space associated the data for the generators of G. Theorem 1 below-our main theorem-is that in the case r = 4 this is equivalent to finding a Q-point on a curve derived from a quotient of the upper half plane by a subgroup of PSLy(l). Although the description of this curve is quite explicit, there is one big problem : while it is sometimes a modular curve (5 4), more often it is not. For this exposition we apply the theory to a simple example that illustrates the main points that arise in the arithmetic of 4 branch point covers (5 5.2 and 5.3). The group is just A. in this case, but this allows us to compare the generalizations of m d @ with the historical progenitor of this, Hilbert's method for realizing alternating groups as Galois groups (5 5.3). Description of the main results. The theory of the arithmetic of covers of the sphere arises in many diophantine investigations. The most well known, of course, is a version of the inverse problem of Galois theory : does every finite group G arise as the group of a Galois extension L/Wx) with fl n L = ((i.e. L/ x) is a regular extension) ? For this lecture we use the dual theory of finite covers rf : X+ IP1 of projective nonsingular curves. We shall consistently assume in this notation that P1 is identified with C U co = IP; , a copy of the complex plane uniformized by x, together with a point at CO. Such a rover corresponding to the field extension L/Q(x) would have the property that it is defined over (((5 1.1) and the induced map on the function field level recovers the field extension L/Q(x). It is valuable, as we shall see in the key example of the paper, to consider covers rf : X + Pi that may not be Galois.