  We study questions around the existence of bounds and the dependence on
parameters for linear-algebraic problems in polynomial rings over rings of an
arithmetic flavor.In particular, we show that the module of syzygies of
polynomials $f_1,...,f_n\in R[X_1,...,X_N]$ with coefficients in a Pr\"ufer
domain $R$ can be generated by elements whose degrees are bounded by a number
only depending on $N$, $n$ and the degree of the $f_j$. This implies that if
$R$ is a B\'ezout domain, then the generators can be parametrized in terms of
the coefficients of $f_1,...,f_n$ using the ring operations and a certain
division function, uniformly in $R$.
