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：多重分级贝蒂数是多参数持续同调模中最简单的不变量之一。该不变量在理论上具有重要价值——它完全确定了模的希尔伯特函数及其最小自由分解中自由模的同构类型——同时在实践中也易于可视化，是当前多参数持续同调软件（如RIVET）的主要输出。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，这种潜在的稳定性缺失限制了其实际应用。我们通过引入一种高效计算的瓶颈型差异函数，证明了多重分级贝蒂数的稳定性结果。受近期关于带符号条码工作的启发，我们的匹配概念允许在相同模的不同奇偶同调次数间进行条码匹配，同时在相同奇偶同调次数的不同模间进行条码匹配。该稳定性结果是希尔伯特合冲定理、Bjerkevik的自由模瓶颈稳定性以及关于射影分解的新稳定性结果的综合。此外，针对双参数情形，我们还证明了希尔伯特函数关于Bjerkevik与Lesnick的1-呈现距离的1-Wasserstein稳定性。