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A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.

Let $$R = K[x_1,\ldots ,x_r]$$ be the polynomial ring over an infinite field K. For a class of Artinian K-algebras $$A = R/I$$ , where I is a monomial ideal of certain specific form and K has some positive characteristics, we examine the weak Lefschetz property of A for various choices of I. In particular, these results support parts of a conjecture by Migliore, Miro-Roig and Nagel in some positive characteristics, and reveal that another part of their conjecture is characteristic-dependent.

We show that a Buchsbaum simplicial complex of small codimension must have large depth. More generally, we achieve a similar result for ${\rm CM}_t$ simplicial complexes, a notion generalizing Buchsbaum-ness, and we prove more precise results in the codimension 2 case. Along the paper, we show that the ${\rm CM}_t$ property is a topological invariant of a simplicial complex., 9 pages, 1 figure

We give a generalization of Eagon-Reiner’s theorem relating Betti numbers of the Stanley-Reisner ideal of a simplicial complex and the CMt property of its Alexander dual. Then we characterize bi-CMt bipartite graphs and bi-CMt chordal graphs. These are generalizations of recent results due to Herzog and Rahimi.

We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen-Macaulay if and only if it is pure and its one dimensional links are connected, and, a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For $t\ge 2$, a lexsegment complex is strictly Cohen-Macaulay in codimension $t$ if and only if it is the join of a lexsegment pure disconnected flag complex with a $(t-2)$-dimensional simplex. When the Stanley-Reisner ideal of a pure lexsegment complex is not quadratic, the complex is Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension. Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai., Comment: 6 pages

For a positive integer k and a non-negative integer t, a class of simplicial complexes, to be denoted by k-CMt, is introduced. This class generalizes two notions for simplicial complexes: being k-Cohen–Macaulay and k-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CMt (=1−CMt) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CMt property under the operation of join of two simplicial complexes. We show that a complex is k-CMt if and only if the links of its non-empty faces are k-CMt−1. We prove that for an integer s≤d, the (d−s−1)-skeleton of a (d−1)-dimensional k-CMt complex is (k+s)-CMt. This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.

Let X be a smooth projective variety of dimension r and π:X → ℙm a generic projection with r + 1 ≤ m ≤ 2r. It is shown that, at any point on X′ = π(X) of multiplicity μ, off a closed subset of the triple locus of codimension four, the depth of the local ring is equal to r − (μ − 1)(m − r − 1). This leads to some improvements on the affirmation of a conjecture of Andreotti–Bombieri–Holm on the weak normality of X′ and a conjecture of Piene on the weak normality of Sing(X′).

In this article, using the local parametric equations of a generic projection π of a smooth projective variety X, at an analytically irreducible singular point y of X′ = π(X), the defining ideals J and J′ of X′ and its singular locus at y are expressed as ideals of maximal and sub-maximal minors of certain Sylvester matrix @. The proof is obtained by a convenient reduction of @ to a “generic pluri-circulant matrix” P and the construction of minimal Grobner bases for the ideal of t-minors of P and for the ideals J and J′. The depth of local rings of X′ and Sing (X′) at y are also computed in terms of the multiplicity at y.

For positive integers n , b 1 ⩽ b 2 ⩽⋯⩽ b n and t ⩽ n , let I t be the transversal monomial ideal generated by square-free monomials (∗) y i 1 j 1 y i 2 j 2 ⋯y i t j t , 1⩽i 1 2 t ⩽n, 1⩽j k ⩽b i k , k=1,…,t, where y ij 's are distinct indeterminates. It is observed that the simplicial complex associated to this ideal is pure shellable if and only if b 1 =⋯= b n =1, but its Alexander dual is always pure and shellable. The simplicial complex admits some weaker shelling which leads to the computation of its Hilbert series. The main result is the construction of the minimal free resolution for the quotient ring of I t . This class of monomial ideals includes the ideals of t -minors of generic pluri-circulant matrices under a change of coordinates. The last family of ideals arise from some specializations of the defining ideals of generic singularities of algebraic varieties.

Let be a pluri-circulant matrix, a concatenation of circulant matrices with entries in a commutative ring. Let be the submatrix of the first rows of We provide some basic “determinantal identities” and prove that every -minor of is a -linear combination of the “basic” -minors. When is generic, we show that the set of basic -minors of and the set of “weakly ordered” maximal minors of both form minimal generating sets for the ideal of -minors of . This implies that the quotient ring by the ideal of -minors of a generic circulant matrix is Cohen-Macaulay. When has two blocks, we show that the weakly ordered maximal minors of and the weakly ordered maximal minors of form minimal Grobner bases for the ideals of maximal and sub-maximal minors of , respectively. The motivation for studying this class of determinantal ideals comes from the study of generic multiple points.

We give a Grobner basis for the ideal of 2-minors of a 2 × n utiatrix of linear forms. The minimal free resolution of such an ideal is obtained in [4] when the corresponding Kronecker-Weierstrass normal form has no iiilpotent blocks. For the general case, using this result, the Grobner basis and the Eliahou-Kervaire resolution for stable monomial ideals, we obtain a free resolution with the expected regularity. For a specialization of the defining ideal of ordinary pinch points, as a special case of these ideals, we provide a minimal free resolution explicitly in terms of certain Koszul complex.

For a finite morphism f : X → Y of smooth varieties such that f maps X birationally onto X′=f(X), the local equations of f are obtained at the double points which are not triple. If C is the conductor of X over X′, and \(D = Sing(X') \subset X',\Delta \subset X\) are the subschemes defined by C, then D and Δ are shown to be complete intersections at these points, provided that C has “the expected” codimension. This leads one to determine the depth of local rings of X′ at these double points. On the other hand, when C is reduced in X, it is proved that X′ is weakly normal at these points, and some global results are given. For the case of affine spaces, the local equations of X′ at these points are computed.

This paper analyses the 'm out of n oblivious transfer', presented at the ACISP 2002 Conference. It is shown that the schemes presented in the paper fail to satisfy the requirements of the oblivious transfer.

Let p(x) be a polynomial of degree n⩾2 with coefficients in a subfield K of the complex numbers. For each natural number m⩾2, let Lm(x) be the m×m lower triangular matrix whose diagonal entries are p(x) and for each j=1,…,m−1, its jth subdiagonal entries are p j (x) j! . For i=1,2, let Lmi)(x) be the matrix obtained from Lm(x) by deleting its first i rows and its last i columns. L1(1)(x)≡1. Then, the function Bm(x)=x−p(x) det (L m−1 (1) (x)) det (L m (1) (x)) defines a fixed-point iteration function having mth order convergence rate for simple roots of p(x). For m=2 and 3, Bm(x) coincides with Newton's and Halley's, respectively. The function Bm(x) is a member of S(m,m+n−2), where for any M⩾m, S(m,M) is the set of all rational iteration functions g(x) ∈ K(x) such that for all roots θ of p(x), then g(x)=θ+∑i=mMγi(x)(θ−x)i, with γi(x) ∈ K(x) and well-defined at any simple root θ. Given g ∈ S(m,M), and a simple root θ of p(x), gi(θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is γ m (θ) = (−1)g m (θ) m! . For Bm(x) we obtain γ m (θ)= (−1) m det (L m+1 (2) (θ)) det (L m (1)(θ)) . If all roots of p(x) are simple, Bm(x) is the unique member of S(m,m + n − 2). By making use of the identity 0 = ∑ i=0 n [p (i) (x) i!] (θ − x) i , we arrive at two recursive formulas for constructing iteration functions within the S(m,M) family. In particular, the family of Bm(x) can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schroder, whose mth order member belongs to S(m,mn), m>2. The iteration functions within S(m,M) can be extended to any arbitrary smooth function f, with the uniform replacement of p(j) with f(j) in g as well as in γm(θ).

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of $t$-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gr\"obner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervair resolution, the Betti numbers of this initial ideal is computed and it is proved that, for some significant values of $t$, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper, naturally arise in the study of generic singularities of algebraic varieties., Comment: 15 pages