Rational varieties over finite fields

Let *f(x*_{1},...,x_{n})
be a polynomial of degree at most *n*
over a finite field *K*. Assume that (a_{1},...,a_{n})
and *(b*_{1},...,b_{n}) 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 *h*_{1}(t),..., h_{n}(t)
such that *a*_{i}=h_{i}(0), b_{i}=h_{i}(1)
for every *i*
and *f(h*_{1}(t),..., h_{n}(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.