Bouvier's Conjecture and Dimension Sequences of Unique Factorization Domains

Abstract

We prove Bouvier's conjecture. More generally, an integer sequence $(a_n{)}_{n\ge 0}$ with $a_0=d\ge 0$ is realized by a unique factorization domain (UFD) $R$ with $\dim R[X_1,\dots ,X_n]=a_n$ for every $n\ge 0$ if and only if $$a_n+1\le a_{n+1}\le a_n+\lfloor \frac{a_n+1}{n+1}\rfloor (n\ge 0)$$ and $a_1\le 2d$ whenever $d\ge 1$.