On invariants of certain symmetric algebras (Q1756476)

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring, where $K$ is a field. Let $\mathfrak{m}=(x_1,\ldots,x_n)$. In the paper under review, the authors study about the symmetric algebra of the first syzygy module, $\mathrm{Syz}_1(\mathfrak{m})$, of $\mathfrak{m}$. Denote this algebra by $\mathrm{Sym}(\mathrm{Syz}_1(\mathfrak{m}))$. The authors prove that the multiplicity of $\mathrm{Sym}(\mathrm{Syz}_1(\mathfrak{m}))$ is equal to $1$ for $n \geq 5$ and equal to $5$ when $n=4$ (Theorem 3.1). The Castelnuovo-Mumford regularity, $\mathrm{reg}(\mathrm{Sym}(\mathrm{Syz}_1(\mathfrak{m})))\leq n-2$ for $n\geq 3$ (Theorem 5.1). The main ingredient of the proofs is the concept of the $s$-sequences introduced by \textit{J. Herzog} et al. [Manuscr. Math. 104, No. 4, 479--501 (2001; Zbl 1058.13011)].