  This paper deals with lattices $(L,\Vert~\Vert)$ over polynomial rings, where
$L$ is a finitely generated module over $k[t]$, the polynomial ring over the
field $k$ in the indeterminate $t$, and $\Vert~\Vert$ is a discrete real-valued
length function on $L\otimes_{k[t]}k(t)$. A reduced basis of $(L,\Vert~\Vert)$
is a basis of $L$ whose vectors attain the successive minima of
$(L,\Vert~\Vert)$. We develop an algorithm which transforms any basis of $L$
into a reduced basis of $(L,\Vert~\Vert)$. By identifying a divisor $D$ of an
algebraic function field with a lattice $(L,\Vert~\Vert)$ over a polynomial
ring, this reduction algorithm can be addressed to the computation of the
Riemann-Roch space of $D$ and the successive minima of $(L,\Vert~\Vert)$,
without the use of any series expansion.
