Rational varieties over finite fields
be a polynomial of degree at most n
over a finite field K. Assume that (a1,...,an)
and (b1,...,bn) are two zeros of f in K.
Is there a whole rational curve of solutions connecting these
two? In other words,
are there rational functions h1(t),..., hn(t)
such that ai=hi(0), bi=hi(1)
for every i
and f(h1(t),..., hn(t)) is identically zero?
The answer to this question (and its generalizations)
solve some conjectures of Colliot-Thélène on
R-equivalence and on the Chow group of zero cycles
over local fields.