Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and the isomorphism type of the free modules in its minimal free resolution -- as well as in practice -- it is easy to visualize and it is one of the main outputs of current multiparameter persistent homology software, such as RIVET. However, to the best of our knowledge, no bottleneck stability result with respect to the interleaving distance has been established for this invariant so far, and this potential lack of stability limits its practical applications. We prove a stability result for multigraded Betti numbers, using an efficiently computable bottleneck-type dissimilarity function we introduce. Our notion of matching is inspired by recent work on signed barcodes, and allows matching bars of the same module in homological degrees of different parity, in addition to matchings bars of different modules in homological degrees of the same parity. Our stability result is a combination of Hilbert's syzygy theorem, Bjerkevik's bottleneck stability for free modules, and a novel stability result for projective resolutions. We also prove, in the $2$-parameter case, a $1$-Wasserstein stability result for Hilbert functions with respect to the $1$-presentation distance of Bjerkevik and Lesnick.

翻译：多分次贝蒂数是多参数持续模最简单的不变量之一。该不变量在理论层面具有重要价值——它完全决定了模的希尔伯特函数及其极小自由分解中自由模的同构类型——同时在实践中也有应用价值：易于可视化，且是RIVET等现有多参数持续同调软件的主要输出之一。然而据我们所知，目前尚未建立该不变量关于交织距离的瓶颈稳定性结果，这种稳定性缺失限制了其实际应用。本文通过引入一种可高效计算的瓶颈型相异度函数，证明了多分次贝蒂数的稳定性结果。受近期关于符号条形码研究的启发，我们提出的匹配概念允许在不同奇偶性同调次数的同一模条形码之间，以及相同奇偶性同调次数的不同模条形码之间进行匹配。该稳定性结果是希尔伯特合冲定理、Bjerkevik关于自由模的瓶颈稳定性以及关于投射分解的新稳定性结论的综合运用。此外，对于双参数情形，我们还证明了希尔伯特函数关于Bjerkevik与Lesnick提出的1-表示距离的1-瓦瑟斯坦稳定性结果。