Rational varieties over finite fields

Let f(x_1,...,x_n) be a polynomial of degree less than or equal to 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.