scholarly journals BERNSTEIN–SATO ROOTS FOR MONOMIAL IDEALS IN POSITIVE CHARACTERISTIC

2020 ◽  
pp. 1-10
Author(s):  
EAMON QUINLAN-GALLEGO
Keyword(s):  
Positive Characteristic ◽  
Monomial Ideals ◽  
Prime Characteristic ◽  
The Ideal

Following the work of Mustaţă and Bitoun, we recently developed a notion of Bernstein–Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein–Sato polynomial. Here, we prove that for monomial ideals the roots of the Bernstein–Sato polynomial (over $\mathbb{C}$ ) agree with the Bernstein–Sato roots of the mod $p$ reductions of the ideal for $p$ large enough. We regard this as evidence that the characteristic- $p$ notion of Bernstein–Sato root is reasonable.

scholarly journals SUBALGEBRAS OF THE POLYNOMIAL ALGEBRA IN POSITIVE CHARACTERISTIC AND THE JACOBIAN

2011 ◽  
Vol 10 (04) ◽  
pp. 605-613
Author(s):  
ALEXEY V. GAVRILOV
Keyword(s):  
Principal Ideal ◽  
Positive Characteristic ◽  
Polynomial Algebra ◽  
Characteristic P ◽  
The Ideal

Let 𝕜 be a field of characteristic p > 0 and R be a subalgebra of 𝕜[X] = 𝕜[x1, …, xn]. Let J(R) be the ideal in 𝕜[X] defined by [Formula: see text]. It is shown that if it is a principal ideal then [Formula: see text], where q = pn(p - 1)/2.

THE RAMIFICATION GROUPS AND DIFFERENT OF A COMPOSITUM OF ARTIN–SCHREIER EXTENSIONS

2010 ◽  
Vol 06 (07) ◽  
pp. 1541-1564 ◽  
Author(s):  
QINGQUAN WU ◽  
RENATE SCHEIDLER
Keyword(s):  
Class Number ◽  
Function Field ◽  
Positive Characteristic ◽  
Field Extension ◽  
Constant Field ◽  
Divisor Class ◽  
Characteristic P ◽  
Similar Nature ◽  
Field Of Positive Characteristic ◽  
The Ideal

Let K be a function field over a perfect constant field of positive characteristic p, and L the compositum of n (degree p) Artin–Schreier extensions of K. Then much of the behavior of the degree pn extension L/K is determined by the behavior of the degree p intermediate extensions M/K. For example, we prove that a place of K totally ramifies/is inert/splits completely in L if and only if it totally ramifies/is inert/splits completely in every M. Examples are provided to show that all possible decompositions are in fact possible; in particular, a place can be inert in a non-cyclic Galois function field extension, which is impossible in the case of a number field. Moreover, we give an explicit closed form description of all the different exponents in L/K in terms of those in all the M/K. Results of a similar nature are given for the genus, the regulator, the ideal class number and the divisor class number. In addition, for the case n = 2, we provide an explicit description of the ramification group filtration of L/K.

scholarly journals On a generalization of test ideals

2004 ◽  
Vol 175 ◽  
pp. 59-74 ◽  
Author(s):  
Nobuo Hara ◽  
Shunsuke Takagi
Keyword(s):  
Prime Characteristic ◽  
Tight Closure ◽  
Test Ideal ◽  
Important Object ◽  
The Ideal

AbstractThe test ideal τ(R) of a ring R of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal τ(at) associated to a given ideal a with rational exponent t ≥ 0. We first prove a key lemma of this paper (Lemma 2.1), which gives a characterization of the ideal τ(at). As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal τ(R). Moreover, we prove an analogue of so-called Skoda’s theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the “modified Briançon-Skoda theorem.”

scholarly journals m-NIL-CLEAN COMPANION MATRICES

2019 ◽  
Vol 35 ◽  
pp. 626-632
Author(s):  
Andrada Cîmpean
Keyword(s):  
Positive Characteristic ◽  
Prime Characteristic ◽  
Characteristic P ◽  
Companion Matrices

Companion matrices over fields of prime characteristic, p, that are sums of two idempotents and a nilpotent are characterized in terms of dimension and trace of such a matrix and of p. Companion matrices over fields of positive characteristic, p, that are sums of m idempotents, m ≥ 2, and a nilpotent are characterized in terms of dimension and trace of such a matrix and of p.

On the reduction numbers of monomial ideals

2019 ◽  
Vol 19 (10) ◽  
pp. 2050201
Author(s):  
Ibrahim Al-Ayyoub
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Upper Bounds ◽  
Integral Closure ◽  
Monomial Ideals ◽  
Explicit Method ◽  
Generating Set ◽  
Ideal Formula ◽  
Reduction Number ◽  
The Ideal

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.

scholarly journals Splittings and Symbolic Powers of Square-free Monomial Ideals

2019 ◽  
Author(s):  
Jonathan Montaño ◽  
Luis Núñez-Betancourt
Keyword(s):  
Linear Optimization ◽  
Monomial Ideals ◽  
Sufficient Condition ◽  
Graded Algebra ◽  
Prime Characteristic ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Symbolic Powers ◽  
Frobenius Map

Abstract We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

scholarly journals Componentwise linear ideals

1999 ◽  
Vol 153 ◽  
pp. 141-153 ◽  
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Homogeneous Polynomials ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Alexander Dual ◽  
Componentwise Linear Ideals ◽  
The Ideal

AbstractA componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal IΔ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

scholarly journals Some Families of Componentwise Linear Monomial Ideals

2007 ◽  
Vol 187 ◽  
pp. 115-156 ◽  
Author(s):  
Christopher A. Francisco ◽  
Adam Van Tuyl
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Fat Points ◽  
Graded Betti Numbers ◽  
Alexander Duality ◽  
Componentwise Linear Ideals ◽  
Positive Integers ◽  
The Ideal

AbstractLet R = k[x1,…,xn] be a polynomial ring over a field k. Let J = {j1,…,jt} be a subset of {1,…, n}, and let mJ ⊂ R denote the ideal (xj1,…,xjt). Given subsets J1,…,Js of {1,…, n} and positive integers a1,…,as, we study ideals of the form These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s ≤ 3 or when Ji ∪ Jj = [n] for all i ≠ j. When s ≥ 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s = 2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k) = 0, our work also yields new cases in which this conjecture holds.

scholarly journals Free resolution of powers of monomial ideals and Golod rings

2017 ◽  
Vol 120 (1) ◽  
pp. 59 ◽  
Author(s):  
N. Altafi ◽  
N. Nemati ◽  
S. A. Seyed Fakhari ◽  
S. Yassemi
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Monomial Order ◽  
Golod Ring ◽  
Linear Quotients ◽  
The Ideal

Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies the gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotients with respect to a monomial order.

scholarly journals Roots of Bernstein-Sato polynomials for monomial ideals: a positive characteristic approach

2006 ◽  
Vol 13 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Nero Budur ◽  
Mircea Mustata ◽  
Morihiko Saito
Keyword(s):  
Positive Characteristic ◽  
Monomial Ideals
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