Rational varieties over finite fields
 <BR><BR>




Let <i>f(x<sub>1</sub>,...,x<sub>n</sub>)</i> 
be a polynomial of degree at most <i>n</i> 
over a finite field <i>K</i>. Assume that </i>(a<sub>1</sub>,...,a<sub>n</sub>)
</i>
and <i>(b<sub>1</sub>,...,b<sub>n</sub>)</i> are two zeros of <i>f</i> in <i>K</i>. 
Is there a whole rational curve of solutions connecting these
two? In other words,
are there rational functions <i>h<sub>1</sub>(t),..., h<sub>n</sub>(t)</i>
such that <i>a<sub>i</sub>=h<sub>i</sub>(0), b<sub>i</sub>=h<sub>i</sub>(1)
</i> for every <i>i</i>
and  <i>f(h<sub>1</sub>(t),..., h<sub>n</sub>(t))</i> is identically zero?
<BR><BR>

The answer to  this question (and its generalizations)
solve some conjectures of Colliot-Th&eacute;l&egrave;ne on
R-equivalence and on the Chow group of zero cycles
over local fields.