2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67802We 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 Prüfer 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 Bézout 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 $R$.36 pagesCommutative AlgebraLogicPrimary 13D02; Secondary 13F05, 13L05text