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Bounds and definability in polynomial rings

Matthias Aschenbrenner
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Abstract

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 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 $R$.

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Matthias Aschenbrenner. 2003-06-16. Bounds and definability in polynomial rings. https://arxiv.org/abs/math/0306240

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