An upper bound for the first Hilbert coefficient of Gorenstein algebras and modules

Let R R be a polynomial ring over a field and M = ⨁ n M n M= \bigoplus _n M_n be a finitely generated graded R R -module, minimally generated by homogeneous elements of degree zero with a graded R R -minimal free resolution F \mathbf {F} . A Cohen-Macaulay module M M is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, e 1 e_1 in terms of the shifts in the graded resolution of M M . When M = R / I M = R/I , a Gorenstein algebra, this bound agrees with the bound obtained in [ES09] in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.

Hilbert stereo reconstruction algorithm based on depth feature and stereo matching

This paper proposes a Hilbert stereo reconstruction algorithm based on depth feature and stereo matching to solve the problem of occlusive region matching errors, namely, the Hilbert stereo network. The traditional stereo network pays more attention to disparity itself, leading to the inaccuracy of disparity estimation. Our design network studies the effective disparity matching and refinement through reconstruction representation of Hilbert’s disparity coefficient. Since the Hilbert coefficient is not affected by the occlusion and texture in the image, stereo disparity matching can conducted effectively. Our network includes three sub-modules, namely, depth feature representation, Hilbert cost volume fusion, and Hilbert refinement reconstruction. Separately, texture features of different depth levels of the image were extracted through Hilbert filtering operation. Next, stereoscopic disparity fusion was performed, and then Hilbert designed to refine the difference regression stereo matching solution was used. Based on the end-to-end design, the structure is refined by combining the depth feature extraction module and Hilbert coefficient disparity. Finally, the Hilbert stereo matching algorithm achieves excellent performance on standard big data set and is compared with other advanced stereo networks. Experiments show that our network has high accuracy and high performance.

The second Hilbert coefficient of modules with almost maximal depth

Let [Formula: see text] be a good [Formula: see text]-filtration of a finitely generated [Formula: see text]-module [Formula: see text] of dimension [Formula: see text], where [Formula: see text] is a local ring and [Formula: see text] is an [Formula: see text]-primary ideal of [Formula: see text]. In the case of depth [Formula: see text], we give an upper bound for the second Hilbert coefficient [Formula: see text] generalizing the results by Huckaba–Marley, and Rossi–Valla proved that [Formula: see text] is Cohen–Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module [Formula: see text]. A lower bound on [Formula: see text] is proved generalizing a result by Rees and Narita.

RELATIVE HILBERT CO-EFFICIENTS

Let (A,${\mathfrak{m}$) be a Cohen–Macaulay local ring of dimensiondand letI⊆Jbe two${\mathfrak{m}$-primary ideals withIa reduction ofJ. Fori= 0,. . .,d, leteiJ(A) (eiI(A)) be theith Hilbert coefficient ofJ(I), respectively. We call the numberci(I,J) =eiJ(A) −eiI(A) theith relative Hilbert coefficient ofJwith respect toI. IfGI(A) is Cohen–Macaulay, thenci(I,J) satisfy various constraints. We also show that vanishing of someci(I,J) has strong implications on depthGJn(A) forn≫ 0.

On the Hilbert Coefficients and Betti Numbers of the Stanley-Reisner Ring of a Matroid Complex

Let S=K[x1,…,xn] be a polynomial ring. Herzog and Zheng conjectured that the i-th Hilbert coefficient of a finitely generated graded Cohen-Macaulay S-module N generated in degree 0 is bounded by the functions of the minimal and maximal shifts in the minimal graded free resolution of N over S and the 0-th Betti number of N. Also, Römer asked whether under the Cohen-Macaulay assumption the i-th Betti number of S/I, where I ⊂ S is a graded ideal, can be bounded by the functions of the minimal and maximal shifts of S/I. In this paper, we provide elementary proofs to establish Herzog and Zheng's conjecture and the upper bound part of Römer's question for the Stanley-Reisner ring of a matroid complex.

Upper bounds of Hilbert coefficients and Hilbert functions

Let (R, m) be a d-dimensional Cohen–Macaulay local ring. In this paper we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a m-primary ideal I ⊂ R that improves all known upper bounds unless for a finite number of cases, see Remark 2.3. We also provide new upper bounds of the Hilbert functions of I extending the known bounds for the maximal ideal.