Birch's theorem in function fields

Abstract

We establish an aysmptotic formula for the number of points with coordinates in $\mb{F}_q[t]$ on a complete intersection of degree $d$ defined over $\mb{F}_q[t]$, with explicit error term, provided that the characteristic of $\mb{F}_q$ is greater than $d$, the codimension of the singular locus of the complete intersection is large enough, and this intersection has a non-singular point at each place of $\mb{F}_q[t]$. In particular, when this complete intersection is non-singular, we show that it satisfies weak approximation.