‎we obtain the porc formulae for the number of non-associative algebras‎ ‎of dimension 2‎, ‎3 and 4 over the finite field gf$(q)$‎. ‎we also give some‎ ‎asymptotic bounds for the number of algebras of dimension $n$ over gf$(q)$.

Serre [5] has recently proved a general theorem about projective modules over commutative rings. This theorem has the following consequence : If 7T is a finite abelian group, any finitely generated projective module over the integral group ring Zir is the direct sum of a free module and an ideal of Zir. The question naturally arises as to whether this result holds for nonabelian groups x. Serre...

‎we obtain the porc formulae for the number of non-associative algebras‎ ‎of dimension 2‎, ‎3 and 4 over the finite field gf$(q)$‎. ‎we also give some‎ ‎asymptotic bounds for the number of algebras of dimension $n$ over gf$(q)$‎.

A smooth geometrically connected curve over the finite field [Formula: see text] with gonality has at most rational points. Faber and Grantham conjectured that there exist curves of every sufficiently large genus achieve this bound. In paper, we show bound can be achieved for an infinite sequence genera using abelian covers projective line. We also argue will not suffice to prove full conjecture.

Given a projective variety X defined over a finite field, the zeta function of divisors counts all irreducible, codimension one subvarieties of X, each measured by their projective degree. For dim(X) ≥ 2, this is a purely p-adic function. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to the all of Cp. When the divisor class group of X has rank one,...

This article continues the characterization of elliptic curves among sets in a finite plane which are met by lines in at most three points. The case treated here is that of sets of prime-power cardinality. 1 Notation GF (q) the finite field of q elements PG(2, q) the projective plane over GF (q) PG(1)(2, q) the set of lines in PG(2, q) P(X) the point of PG(2, q) with coordinate vector X PQ the ...

Tsfasman-Boguslavsky Conjecture predicts the maximum number of zeros that a system of linearly independent homogeneous polynomials of the same positive degree with coefficients in a finite field can have in the corresponding projective space. We give a self-contained proof to show that this conjecture holds in the affirmative in the case of systems of three homogeneous polynomials, and also to ...

1.1 Projective spaces and spreads. A projective space is a geometry consisting of a set of points and a set of lines, where each line is a subset of the point set, such that the following axioms hold: • Any two points are on exactly one line. • Let A, B, C, D be four distinct points no three of which are collinear. If the lines AB and CD intersect each other, then the lines AD and BC also inter...

A basis of the ideal of the complement of a linear subspace in a projective space over a finite field is given. As an application, the second largest number of points of plane curves of degree d over the finite field of q elements is also given for d ≥ q + 1.