Fast computation of a rational point of a variety over a finite field

Details Fast computation of a rational point of a variety over a finite field Fast computation of a rational point of a variety over a finite field

2004-06-04

arxiv.org/abs/math/0406085v1

Mathematics

Number Theory

36 pages

Scientific paper

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.

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