Géza Kós

Reconstruction of matrices from submatrices

For an arbitrary matrix A of n x n symbols, consider its submatrices of size k x k, obtained by deleting n―k rows and n―k columns. Optionally, the deleted rows and columns can be selected symmetrically or independently. We consider the problem of whether these multisets determine matrix A. Following the ideas of Krasikov and Roditty in the reconstruction of sequences from subsequences, we replace the multiset by the sum of submatrices. For k > cn 2/3 we prove that the matrix A is determined by the sum of the k x k submatrices, both in the symmetric and in the nonsymmetric cases.

Some extensions of Alon's Nullstellensatz

Alons combinatorial Nullstellensatz, and in particular the resulting nonvanishing criterion is one of the most powerful algebraic tools in combinatorics, with many important applications. In this paper we extend the nonvanishing theorem in two directions. We prove a version allowing multiple points. Also, we establish a variant which is valid over arbitrary commutative rings, not merely over subrings of fields. As an application, we prove extensions of the theorem of Alon and Furedi on hyperplane coverings of discrete cubes.

Simultaneous extensions of turkevich's inequality and the weighted AM-GM inequality

We establish a sharp homogeneous inequality which extends both the classical weighted AM-GM inequality and the Turkevich inequality.