A Characterization of extendibility of rational matrices is presented in terms of elementary properties. As a tool we give a solvability condition for a system of linear diophantine equations, which is of independent interest.
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The property of extendibility of rational matrices was introduced by Sivakumar [3] in his investigation of linear independence of integer translates of exponential box splines with rational directions. This property was subsequently extended and refined by Ron [2]. The purpose of this note is to provide a characterization of extendibility in terms of elementary properties.
Definition. Let Y/Q s be a linearly independent set of 1 k s vectors. We say that Y is extendible if there is a matrix X s_s with an integral inverse whose first k columns constitute Y. For an arbitrary s_n matrix 5, we say that 5 is fully extendible if every linearly independent subset Y of 5 is extendible.
Note that any s_n rational matrix can be written as (1ΓP) 5 with P # N and 5 # Z s_n , which is crucial in our investigation of box splines with rational directions [5]. So in what follows we always take such a form for a rational matrix. For an l_m integer matrix A we also think of it as the multiset of its column vectors and denote *A as its cardinality. Also define d A, r as the greatest common divisor of all r_r minors of A. Set d A, 0 =1. Then our main result can be stated as follows.