As part of the graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because existential, provides no explicit information about these excluded minors. In 1993, Seymour established the first upper bound on the order of such minimal excluded minors. Very recently, Houdaigoui and Kawarabayashi improved this result by deriving a quasi-polynomial upper bound. Despite this progress, the gap between this bound and the known linear lower bound $Ω(g)$ (where $g$ denotes the genus) remains substantial. In particular, they conjectured that a polynomial upper bound should hold. In this paper, we confirm this conjecture by showing that the order of the minimal excluded minors for a surface of genus $g$ is $O(g^{8+\varepsilon})$ for every $\varepsilon >0$. This result significantly narrows the gap between the known lower and upper bounds, bringing the asymptotic behavior much closer to the conjectured optimum. Our approach introduces a new forbidden structure of minimal excluded minors. Let $G$ be a minimal excluded minor for a surface of Euler genus $g$. Houdaigoui and Kawarabayashi showed that $G$ contains $O(\log g)$ pairwise disjoint cycles that are contractible and nested in some embedding of $G$. We strengthen this result by proving a separator-based variant: for any contractible subgraph $H \subseteq G$ with a separator of size $s$ (with $H$ completely contained in one side), the subgraph $H$ contains $O(\log s)$ disjoint cycles that are contractible and nested in some embedding of $G$. This allows us to replace a genus-dependent bound with a separator-dependent one, which is the main new ingredient in deriving our polynomial bound.

翻译：作为图子式项目的一部分，Robertson和Seymour于1990年表明，可嵌入给定曲面的图类可由有限个极小禁止子式刻画。然而，其证明由于存在性，未提供关于这些禁止子式的任何显式信息。1993年，Seymour建立了此类极小禁止子式阶数的首个上界。近期，Houdaigoui和Kawarabayashi通过推导拟多项式上界改进了这一结果。尽管取得进展，该上界与已知线性下界$Ω(g)$（其中$g$表示亏格）之间的差距仍然显著。特别地，他们猜想多项式上界应成立。本文通过证明对任意$\varepsilon >0$，亏格$g$曲面的极小禁止子式阶数为$O(g^{8+\varepsilon})$，证实了这一猜想。该结果显著缩小了已知下界与上界之间的差距，使渐近行为更接近猜想的最优值。我们的方法引入了极小禁止子式的一种新禁止结构。设$G$为欧拉亏格$g$曲面的极小禁止子式。Houdaigoui和Kawarabayashi证明了$G$包含$O(\log g)$个两两不交的环，这些环在$G$的某个嵌入中可缩且嵌套。我们通过证明基于分离器的变体来强化该结果：对于任意可缩子图$H \subseteq G$（其分离器大小为$s$，且$H$完全位于一侧），子图$H$包含$O(\log s)$个在$G$的某个嵌入中可缩且嵌套的不交环。这使得我们能用基于分离器的界替代依赖于亏格的界，这是推导多项式界的主要新要素。