Issue 3, Volume 49, September 2024

2024

1. Preface Preface PDF

   309-309
2. Vanishing and Non-negativity of the First Normal Hilbert Coefficient Linquan Ma , Pham Hung Quy PDF

   ▸ Abstract

   Let $(R,\mathfrak {m})$ be a Noetherian local ring such that $\widehat{R}$ is reduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists a parameter ideal $Q\subseteq R$ such that $\bar{e}_1(Q)=0$, then $R$ is regular and $\nu (\mathfrak {m}/Q)\le 1$. This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal [Goto, S., Hong, J., Mandal, M.: The positivity of the first coefficients of normal Hilbert polynomials. Proc. Amer. Math. Soc. 139(7), 2399–2406 (2011)]. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if $\widehat{R}$ is equidimensional, then $\bar{e}_1(Q)\ge 0$ for all parameter ideals $Q\subseteq R$, and in characteristic $p>0$, we actually have $e_1^*(Q)\ge 0$. Our proofs rely on the existence of big Cohen-Macaulay algebras.

   311-325
3. Normalization of Singular Contact Forms and Primitive 1-forms Kai Jiang , Hong Minh Truong , Nguyen Tien Zung PDF

   ▸ Abstract

   A differential 1-form $\alpha$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $\alpha \wedge (d\alpha )^n \ne 0$ almost everywhere, but which vanishes at a point $O$, i.e., $\alpha(O)=0$, is called a singular contact form at $O$. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form $\omega$ in dimension $2n$, i.e., differential 1-forms $\gamma$ which vanish at a point and such that $d\gamma=\omega$, and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin (1975), Webster (Amer. J. Math. 109, 807–832 (1987)) and Zhitomirskii (1986, 1992). Besides the classical normalization techniques, such as the step-by step normalization methods based on the cohomological equations and the Moser path method, we also use the toric approach to the normalization problem for dynamical systems (Jiang et al. 2019; Zung, Ann. Math. 161, 141–156 2005; Zung 2016; Zung, Arch. Rational Mech. Anal. 229, 789–833 (2018)).

   327-345
4. Betti Numbers of the Tangent Cones of Monomial Space Curves Nguyen P. H. Lan , Nguyen Chanh Tu , Thanh Vu PDF

   ▸ Abstract

   Let $H = \langle n_1, n_2,n_3\rangle$ be a numerical semigroup. Let $\widetilde{H}$ be the interval completion of $H$, namely the semigroup generated by the interval $\langle n_1,n_1+1, \ldots , n_3\rangle$. Let $K$ be a field and $K[H]$ the semigroup ring generated by $H$. Let $I_H^*$ be the defining ideal of the tangent cone of $K[H]$. In this paper, we describe the defining equations of $I_H^*$. From that, we prove the Herzog-Stamate conjecture for monomial space curves stating that $\beta _i(I_H^{*}) \le \beta _i(I_{\widetilde{H}}^*)$ for all $i$, where $\beta _i(I_H^*)$ and $\beta _i(I_{\widetilde{H}}^*)$ are the $i$th Betti numbers of $I_H^{*}$ and $I_{\widetilde{H}}^*$ respectively.

   347-365
5. On the Index of Depth Stability of Symbolic Powers of Cover Ideals of Graphs S. A. Seyed Fakhari , S. Yassemi PDF

   ▸ Abstract

   Let $G$ be a graph with $n$ vertices and let $S=\mathbb {K}[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb {K}$. Assume that $I(G)$ and $J(G)$ denote the edge ideal and the cover ideal of $G$, respectively. We provide a combinatorial upper bound for the index of depth stability of symbolic powers of $J(G)$. As a consequence, we compute the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile, we determine a class of graphs $G$ with the property that the Castelnuovo–Mumford regularity of $S/I(G)$ is equal to the induced matching number of $G$.

   367-376
6. Sparse-grid Sampling Recovery and Numerical Integration of Functions Having Mixed Smoothness Dinh Dũng PDF

   ▸ Abstract

   We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.

   377-426
7. On the Diagonal Subgroup of the Special Linear Group Over a Division Ring Bui Xuan Hai PDF

   ▸ Abstract

   Let $K$ be a division ring with center $Z(K)$, and $n$ a positive integer. Let $\textrm{SL}(n,K)$ be the special linear group of degree $n$ over $K$ and $\textrm{SD}(n,K)$ its subgroup consisting of all diagonal matrices whose Dieudonne’s determinant is $\overline{1}$. We prove that $\textrm{SD}(n,K)$ is weakly pronormal, but not pronormal in $\textrm{SL}(n,K)$ provided either $Z(K)$ is an infinite field in case $n\ge 3$ or $Z(K)$ is a finite field containing at least seven elements in case $n\ge 5$.

   427-440
8. Generalized Semi-infinite Polynomial Optimization and Semidefinite Programming Relaxations Liguo Jiao , Jae Hyoung Lee , Tiến-Sơn Phạm PDF

   ▸ Abstract

   This paper focuses on the study of a generalized semi-infinite programming, where the objective and the constraint functions are all real polynomials. We present a method for finding its global minimizers and global minimum using a hierarchy of semidefinite programming relaxations and prove the convergence result for the method. Numerical experiments are presented to show the efficiency of the proposed algorithm.

   441-457
9. Minimal Balanced Neighborly Polynomials Satoshi Murai , Nguyen Thi Thanh Tam PDF

   ▸ Abstract

   In this paper we introduce minimal balanced neighborly polynomials and show some methods to construct such polynomials. In particular, using this notion, we prove the existence of balanced neighborly polynomials of the following types: (i) type $(p,\dots ,p)$ for most prime numbers $p$, (ii) types $(d-1,d,d,d)$, $(d-1,d-1,d,d)$ and $(d-1,d-1,d-1,d)$ when d is odd or is divisible by 4. We also construct balanced neighborly simplicial spheres of type $(2,4k-1,4k-1,4k-1)$.

   459-484
10. Note on the Linear Independence of Alternating Multiple Zeta Values in Positive Characteristic Bo-Hae Im , Hojin Kim , Khac Nhuan Le , Tuan Ngo Dac , Lan Huong Pham PDF

    ▸ Abstract

    We discuss certain results related to the linear independence of alternating multiple zeta values introduced by Harada in 2021.

    485-521
11. The Weak Lefschetz Property of Artinian Algebras Associated to Paths and Cycles Hop D. Nguyen , Quang Hoa Tran PDF

    ▸ Abstract

    Given a base field $\Bbbk$ of characteristic zero, for each graph $G$, we associate the artinian algebra $A(G)$ defined by the edge ideal of $G$ and the squares of the variables. We study the weak Lefschetz property of $A(G)$. We classify some classes of graphs with relatively few edges, including paths and cycles, such that its associated artinian ring has the weak Lefschetz property.

    523-544
12. Refinements and Extensions of Some Strong Duality Theorems in Conic Linear Programming Nguyen Ngoc Luan , Nguyen Dong Yen PDF

    ▸ Abstract

    In this paper, we establish a series of new results on strong duality and solution existence for conic linear programs in locally convex Hausdorff topological vector spaces and finite-dimensional Euclidean spaces. Namely, under certain regularity conditions based on quasi-relative interiors of convex sets, we prove that if one problem in the dual pair consisting of a primal program and its dual has a solution, then the other problem also has a solution, and the optimal values of the problems are equal. In addition, we show that if the cones are generalized polyhedral convex, then the regularity conditions can be omitted. Moreover, if the spaces are finite-dimensional and the ordering cones are closed convex, then instead of the solution existence condition, it suffices to require the finiteness of the optimal value. The present paper complements our recent research work [Luan, N.N., Yen, N.D.: Strong duality and solution existence under minimal assumptions in conic linear programming. J. Optim. Theory Appl. (https://doi.org/10.1007/s10957-023-02318-w)].

    545-561