Bounds and definability in polynomial rings

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 $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n\in R[X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_N]$ 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 $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n$ using the ring operations and a certain division function, uniformly in $R$.