We show that in a polynomial ring $R$ in $N$ variables over an algebraically closed field $K$ of arbitrary characteristic, any $K$-subalgebra of $R$ generated over $K$ by at most $n$ forms of degree at most $d$ is contained in a $K$-subalgebra of $R$ generated by $B \leq {}^\eta\mathcal{B}(n,d)$ forms $G_1,..., G_B$ of degree $\leq d$, where ${}^\eta\mathcal{B}(n,d)$ does not depend on $N$ or $K$, such that these forms are a regular sequence and such that for any ideal $J$ generated by forms that are in the $K$-span of $G_1, ..., G_B$, the ring $R/J$ satisfies the Serre condition $R_\eta$. These results imply a conjecture of M. Stillman asserting that the projective dimension of an $n$-generator ideal $I$ of $R$ whose generators are forms of degree $\leq d$ is bounded independent of $N$. We also show that there is a primary decomposition of $I$ such that all numerical invariants of the decomposition (e.g., the number of primary components and the degrees and numbers of generators of all of the prime and primary ideals occurring) are bounded independent of $N$.

中文翻译：

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多项式环的小子代数和斯蒂尔曼猜想

我们表明，在具有任意特征的代数闭域 $K$ 上的 $N$ 变量中的多项式环 $R$ 中，$R$ 的任何 $K$-子代数在 $K$ 上生成的至多 $n$ 形式至多 $d$ 的度数包含在由 $B \leq {}^\eta\mathcal{B}(n,d)$ 形式 $G_1,..., G_B 生成的 $R$ 的 $K$-子代数中度数 $\leq d$，其中 ${}^\eta\mathcal{B}(n,d)$ 不依赖于 $N$ 或 $K$，因此这些形式是一个规则序列，并且使得对于由 $G_1, ..., G_B$ 的 $K$-span 中的形式生成的任何理想 $J$，环 $R/J$ 满足 Serre 条件 $R_\eta$。这些结果意味着 M. Stillman 的一个猜想，断言 $R$ 的 $n$-生成器理想 $I$ 的射影维数是有界的，其生成器是度数 $\leq d$ 的形式，与 $N$ 无关。