Rational varieties over finite fields

Let f(x1,...,xn) 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.