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 $\dim R[X_1,\ldots,X_n]=a_n$ for every $n \ge 0$ if and only if $$ a_n+1\le a_{n+1} \le a_n+\left\lfloor\frac{a_n+1}{n+1}\right\rfloor \qquad(n\ge0) $$ and $a_1 \le 2d$ whenever $d \ge 1$.