Zero-sum Square Matrices

Authors: Balister P.¹; Caro Y.²; Rousseau C.³; Yuster R.⁴

Source: European Journal of Combinatorics, Volume 23, Number 5, July 2002 , pp. 489-497(9)

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Abstract:

Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-summodp if the sum of each row of B and the sum of each column of B is a multiple of p. Let M(p, k) denote the least integer m for which every square matrix of order at least m has a square submatrix of order k which is zero-sum modp. In this paper we supply upper and lower bounds for M(p, k). In particular, we prove that limsupM(2, k) / k 4, liminfM(3, k) / k 20, and that M(p, k) k22 eexp(1 / e)^{p / 2}. Some nontrivial explicit values are also computed. Copyright 2002 Elsevier Science Ltd.

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240, U.S.A. 2: Department of Mathematics, University of Haifa-ORANIM, Tivon 36006, Israel 3: Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240, U.S.A. 4: Department of Mathematics, University of Haifa-ORANIM, Tivon 36006, Israel

Publication date: 2002-07-01