Remarks on the Hilbert depth of powers of the maximal graded ideal

Mircea Cimpoeaș, Silviu Bălănescu

Abstract


Let $\mathbf m=(x_1,\ldots,x_n)$ be the maximal graded ideal of $S:=K[x_1,\ldots,x_n]$. We present a new method for computing the Hilbert depth of $\mathbf m^t$, using the polarization and a combinatorial characterization of the Hilbert depth of a quotient of squarefree monomial ideals.


Full Text:

PDF

References


J. Apel, On a conjecture of R. P. Stanley; Part II - Quotients Modulo Monomial Ideals, J. of Alg. Comb. 17 (2003), 57–74.

C. Biro, D. M. Howard, M. T. Keller, W. T. Trotter, S. J. Young, Interval partitions and Stanley depth, Journal of Combinatorial Theory Series A 117 (2010), no. 4, 475–482.

W. Bruns, C. Krattenthaler, J. Uliczka, Hilbert depth of powers of the maximal ideal, Contemp. Math. 555 (2011), 1–12.

S. Balanescu, M. Cimpoeas, C. Krattenthaller, On the Hilbert depth of monomial ideals, (2024), https://arxiv.org/pdf/2306.09450.pdf

M. Cimpoeas, Some remark on the Stanley depth for multigraded modules, Le Matematiche Vol. LXII (2008), nr. 2, 165–171.

A. M. Duval, B. Goeckneker, C. J. Klivans, J. L. Martine, A non-partitionable Cohen-Macaulay simplicial complex, Advances in Mathematics 299 (2016), 381–395.

J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, Journal of Algebra 322 (2009), no. 9, 3151–3169.

J. Herzog, A survey on Stanley depth, In: (A.M. Bigatti, P. Gimenez, E. Saenz-de-Cabezon (Eds.)) Monomial Ideals, Computations and Applications, Springer, 2013, 3–45.

R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), 175–193.




DOI: https://doi.org/10.52846/ami.v51i2.1911