Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k–rational point x0. Let ̟(M,x0) denote the corresponding fundamental group–scheme introduced by Nori. Let EG be a principal G–bundle over M , where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M . We prove that the following thr...

A cap of size and degree in a projective space, (briefly; -cap) is set points with the property that each line space meet it at most points. The aim this research to extend complete caps incomplete caps, -caps finite dimension three over field order eleven, which already exist founded by action subgroups general linear group eleven four, -complete caps. These have been classified giving -distri...

(Algebra) It’s a complex projective manifold C of dimension one. As such, it has a field of rational functions K(C) of transcendence degree one over the field C of scalars. The curve can be recovered from the field as the set of discrete valuation rings of K/C, which has the “natural” structure of a projective manifold. There is a canonical line bundle ωC , whose degree captures the genus via d...

Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over with three points removed. We investigate which finite groups $G$ can arise as monodromy group \'{e}tale covers that are tamely ramified removed points. This provides new information about tame fundamental line. In particular, we show for each prime $p\ge 5$, there families symmetric $S_n$ or alt...

Define the height function h(a) = min{k + (ka mod p) : k = 1, 2, . . . , p − 1} for a ∈ {0, 1, . . . , p − 1.} It is proved that the height has peaks at p, (p+1)/2, and (p+c)/3, that these peaks occur at a = [p/3], (p−3)/2, (p− 1)/2, [2p/3], p − 3, p− 2, and p − 1, and that h(a) ≤ p/3 for all other values of a. 1. Heights on finite projective spaces Let p be an odd prime and let Fp = Z/pZ and F...

We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.

Let S be a subset of Fq, the field of q elements and h ∈ Fq[x] a polynomial of degree d > 1 with no roots in S. Consider the group generated by the image of {x − s | s ∈ S} in the group of units of the ring Fq[x]/(h). In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]...

A permutation rational function f ∈ Fq(x) is a rational function that induces a bijection on Fq, that is, for all y ∈ Fq there exists exactly one x ∈ Fq such that f(x) = y. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show ho...

Let S be a subset of Fq, the field of q elements and h ∈ Fq[x] a polynomial of degree d > 1 with no roots in S. Consider the group generated by the image of {x − s | s ∈ S} in the group of units of the ring Fq[x]/(h). In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]...

In the present paper, we give necessary and sufficient conditions for a birational Galois section of a projective smooth curve over either the field of rational numbers or an imaginary quadratic field to be geometric. As a consequence, we prove that, over such a small number field, to prove the birational section conjecture for projective smooth curves, it suffices to verify that, roughly speak...