  We exhibit a probabilistic algorithm which computes a rational point of an
absolutely irreducible variety over a finite field defined by a reduced regular
sequence. Its time--space complexity is roughly quadratic in the logarithm of
the cardinality of the field and a geometric invariant of the input system
(called its degree), which is always bounded by the Bezout number of the
system. Our algorithm works for fields of any characteristic, but requires the
cardinality of the field to be greater than a quantity which is roughly the
fourth power of the degree of the input variety.
