Asymptotics of the complexity of systems of integer linear forms for additive computations

Abstract

The computational complexity of integer linear forms is studied. By l2(A) we denote the minimal number of the additions and subtractions required for computing the system of p linear forms in q variables x1, x2, …, xq that are defined by an integer matrix A of size p × q (repeated use of the results of intermediate computation is permitted). We show that l2(A) ⩾ log D(A), where D(A) is the maximum of the absolute values of the minors of A over all minors from order 1 to order min (p, q) (Theorem 1). Moreover, for each sequence of matrices A(n) of size p(n) × q(n) satisfying the condition p + q = o ((log log D(A))1/2) as n → ∞ the bound l2(A) ⩽ log D(A) + o(log D(A)) is valid (Theorem 2). Hence, for all fixed (and even weakly increasing) sizes of matrices that determine a system of integer linear forms, the upper bound on the computational complexity of this system is asymptotically equal to the lower bound.