Related papers: Non-vanishing forms in projective space over finit…

We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.

We study the projections in vector spaces over finite fields. We prove finite fields analogues of the bounds on the dimensions of the exceptional sets for Euclidean projection mapping. We provide examples which do not have exceptional…

A slip on a paper concerning near-vector spaces is fixed. New characterization of near-vector spaces determined by finite fields is provided and the number (up to the isomorphism) of these spaces is exhibited.

In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of…

We apply the semidefinite programming method to derive bounds for projective codes over a finite field.

We establish sufficient conditions for finite convergence of the alternating projections method for two non-intersecting and potentially nonconvex sets. Our results are based on a generalization of the concept of intrinsic transversality,…

In the previous paper, we established an elementary bound for numbers of points of surfaces in the projective $3$-space over ${\Bbb F}_q$. In this paper, we give the complete list of surfaces that attain the elementary bound. Precisely…

This article is a continuation of a previous article which concerned the splitting problem for subspaces of superspaces. We begin with a general account of projective superspaces. Subsequently, we specialise to subvarieties of `positive'…

For the minimal graph defined on a convex ring in the space form with nonnegative curvature, we obtain the regularity and the strict convexity about its level sets by the continuity method.

Within the framework of finite element systems, we show how spaces of differential forms may be constructed, in such a way that they are equipped with commuting interpolators and contain prescribed functions, and are minimal under these…

In this article, we will introduce methods of non-standard analysis into projective geometry. Especially, we will analyze the properties of a projective space over a non-Archimedean field. Non-Archimedean fields contain numbers that are…

We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with…

We establish an upper bound for the sectional genus of varieties which are invariant under Pfaff fields on projective spaces.

We discuss an elementary, yet unsolved, problem of Niederreiter concerning the enumeration of a class of subspaces of finite dimensional vector spaces over finite fields. A short and self-contained account of some recent progress on this…

We classify fields having finitely many finite non-commutative (not necessarily central) division algebras over them. In the process, we introduce the notion of anti-closure of a field and also make comments on fields having a linear…

We study varieties defined over nonstandard fields using techniques of nonstandard mathematics.

We consider the dimensions of finite type of representations of a partially ordered set, i.e. such that there is only finitely many isomorphism classes of representations of this dimension. We give a criterion for a dimension to be of…

We give an expository account of Nullstellensatz-like results when the base field is finite. In particular, we discuss the vanishing ideal of the affine space and of the projective space over a finite field. As an application, we include an…

We establish that smooth, geometrically integral projective varieties of small degree are not pointless in suitable solvable extensions of their field of definition, provided that this field is algebraic over $\Bbb Q$.

The aim of this paper is to complete the local classification of minimal hypersurfaces with vanishing Gauss-Kronecker curvature in a 4-dimensional space form. Moreover, we give a classification of complete minimal hypersurfaces with…