# Deterministic unimodularity certification

- 2012

#### Abstract

The asymptotically fastest algorithms for many linear algebra problems on integer matrices, including solving a system of linear equations and computing the determinant, use high-order lifting. Currently, high-order lifting requires the use of a randomized shifted number system to detect and avoid error-producing carries. By interleaving quadratic and linear lifting, we devise a new algorithm for high-order lifting that allows us to work in the usual symmetric range modulo <i>p</i>, thus avoiding randomization. As an application, we give a deterministic algorithm to assay if an <i>n</i> x <i>n</i> integer matrix <i>A</i> is unimodular. The cost of the algorithm is <i>O</i>((log <i>n</i>)<i>n</i><sup>ω</sup> M(log <i>n</i> + log||<i>A</i>||)) bit operations, where ||<i>A</i>|| denotes the largest entry in absolute value, and M(<i>t</i>) is the cost of multiplying two integers bounded in bit length by <i>t</i>.

**DOI:**10.1145/2442829.2442870

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