Rational solutions of singular linear systems

A deterministic algorithm is presented for computing a particular solution to a linear system of equations with polynomial coefficients. Given an <italic>A</italic> ∈ <italic>F</italic>[<italic>x</italic>]<supscrpt><italic>n</italic> × <italic>m</italic></supscrpt> and <italic>b</italic> ∈ <italic>F</italic>[<italic>x</italic>]<supscrpt><italic>n</italic></supscrpt>, where <italic>F</italic> is a field, the algorithm will either return a particular solution <italic>v</italic> ∈ <italic>F</italic>(<italic>x</italic>)<supscrpt><italic>m</italic></supscrpt> to the system <italic>Av</italic> = <italic>b</italic> or determine that the system is inconsistent. The cost of the algorithm is <italic>O</italic>((<italic>n</italic> + <italic>m</italic>)<italic>r</italic><supscrpt>2</supscrpt><italic>d</italic><supscrpt>1 + <italic>ε</italic></supscrpt>) field operations from <italic>F</italic>, where <italic>r</italic> is the rank of <italic>A</italic> and <italic>d</italic> - 1 is a bound for the degrees of entries in <italic>A</italic> and <italic>b</italic>.