Reza Akhtar

Zero-Cycles on Varieties Over Finite Fields

Abstract We prove that for a smooth projective variety X of dimension d defined over a finite field k, the structure map σ : X → Spec k induces an isomorphism σ∗ : CH d+1(X, 1) ≅ CH 1(k, 1) = k*. We also prove that the higher Chow groups CH d+s−1(X, s) of one-dimensional cycles are torsion for s ≥ 2.

Representation Numbers of Stars

Abstract A graph G has a representation modulo r if there exists an injective map ƒ : V(G) → {0, 1, . . . , r – 1} such that vertices u and 𝑣 are adjacent if and only if ƒ(u) – ƒ(𝑣) is relatively prime to r. The representation number rep(G) is the smallest positive integer r for which G has a representation modulo r. In this paper we study representation numbers of the stars K 1,n . We will show that the problem of determining rep(K 1,n ) is equivalent to determining the smallest even k for which φ(k) ≥ n: we will solve this problem for “small” n and determine the possible forms of rep(K 1,n ) for sufficiently large n.

Representation numbers of complete multipartite graphs

A graph G has a representation modulo r if there exists an injective map f:V(G)->{0,1,...,r-1} such that vertices u and v are adjacent if and only if f(u)-f(v) is relatively prime to r. The representation number rep(G) is the smallest r such that G has a representation modulo r. Following earlier work on stars, we study representation numbers of complete bipartite graphs and more generally complete multipartite graphs.

Künneth decompositions for quotient varieties

Abstract In this paper we discuss Kunneth decompositions for finite quotients of several classes of smooth projective varieties. The main result is the existence of an explicit (and readily computable) Chow-Kunneth decomposition in the sense of Murre with several pleasant properties for finite quotients of abelian varieties. This applies in particular to symmetric products of abelian varieties and also to certain smooth quotients in positive characteristics which are known to be not abelian varieties, examples of which were considered by Enriques and Igusa. We also consider briefly a strong Kunneth decomposition for finite quotients of projective smooth linear varieties.

Toric residue codes: I

In this paper, we begin exploring the construction of algebraic codes from toric varieties using toric residues. Though algebraic codes have been constructed from toric varieties, they have all been evaluation codes, where one evaluates the sections of a line bundle at a collection of rational points. In the present paper, instead of evaluating sections of a line bundle at rational points, we compute the residues of differential forms at these points. We show that this method produces codes that are close to the dual of those produced by the first technique. We conclude by studying several examples, and also discussing applications of this technique to the construction of quantum stabilizer codes and also to decryption of toric evaluation codes.

On countable homogeneous 3-hypergraphs

SummaryWe present some results on countable homogeneous 3-hypergraphs. In particular, we show that there is no unexpected homogeneous 3-hypergraph determined by a single constraint.

The representation number of some sparse graphs

Abstract We study the representation number for some special sparse graphs. For graphs with a single edge and for complete binary trees we give an exact formula, and for hypercubes we improve the known lower bound. We also study the prime factorization of the representation number of graphs with one edge.

Cycles on curves over global fields of positive characteristic

Let k be a global field of positive characteristic, and let σ: X → Spec k be a smooth projective curve. We study the zero-dimensional cycle group V(X) = Ker(σ*: SIK 1 (X) → K 1 (k)) and the one-dimensional cycle group W(X) = coker(σ*: K 2 (k) → H 0 Zar (X, K 2 )), addressing the conjecture that V(X) is torsion and W(X) is finitely generated. The main idea is to use Abhyankars Theorem on resolution of singularities to relate the study of these cycle groups to that of the K-groups of a certain smooth projective surface over a finite field.

Torsion in Mixed K -Groups

Abstract We define generalized Milnor K-groups known as mixed K-groups and give explicit descriptions of these groups in simple cases. Our main result is a generalization of the theorem of Bass and Tate that the Milnor K-groups (k) are uniquely divisible when k is algebraically closed and n ≥ 2.

The linear chromatic number of a Sperner family

Abstract Let S be a finite set and S a complete Sperner family on S , i.e. a Sperner family such that every x ∈ S is contained in some member of S . The linear chromatic number of S , defined by Civan, is the smallest integer n with the property that there exists a function f : S → { 1 , … , n } such that if f ( x ) = f ( y ) , then every set in S which contains x also contains y or every set in S which contains y also contains x . We give an explicit formula for the number of complete Sperner families on S of linear chromatic number 2 . We also prove tight bounds on the number of elements in a Sperner family of given chromatic number, and prove that complete Sperner families of maximum linear chromatic number are far more numerous those of lesser linear chromatic number.