Exact solution of linear equations usingP-adic expansions

Summary

A method is described for computing the exact rational solution to a regular systemAx=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) ofA for some primep; (ii) using successive refinements to compute an integer vector$\bar x$ such that$A\bar x \equiv b$ (modp ^m) for a suitably large integerm; and (iii) deducing the rational solutionx from thep-adic approximation$\bar x$. For matricesA andb with entries of bounded size and dimensionsn×n andn×1, this method can be implemented in timeO(n ³(logn)²) which is better than methods previously used.