3 problems

Let $k$ be an infinite field and $(R,\mathfrak{m})=k[[x_0,\ldots,x_d]]$. For general linear forms $\ell_1,\ldots,\ell_i\in\mathfrak{m}$, set … Let $\stirlingI{d+1}{d+1-i}$ denote t…

Stirling-number Lech conjecture. The first displayed inequality, together with the equivalent reindexing given in the source, holds for every $\mathfrak{m}$-primary monomial ideal…

Let $(R,\mathfrak{m})$ be a local ring. Suppose its completion $\widehat{R}$ has an isolated singularity, meaning that $\widehat{R}_P$ is regular for every…