EXPONENT MATRICES AND TILED ORDERS OVER DISCRETE VALUATION RINGS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 997-1012 ◽  
Author(s):  
V. V. KIRICHENKO ◽  
A. V. ZELENSKY ◽  
V. N. ZHURAVLEV
Keyword(s):  
Markov Chains ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Strongly Connected ◽  
Exponent Matrix ◽  
Finite Posets

Exponent matrices appear in the theory of tiled orders over a discrete valuation ring. Many properties of such an order and its quiver are fully determined by its exponent matrix. We prove that an arbitrary strongly connected simply laced quiver with a loop in every vertex is realized as the quiver of a reduced exponent matrix. The relations between exponent matrices and finite posets, Markov chains, and doubly stochastic matrices are discussed.

Polynomial index in discrete valuation rings

2019 ◽  
Vol 56 (2) ◽  
pp. 260-266
Author(s):  
Mohamed E. Charkani ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Valuation Ring ◽  
Irreducible Polynomial ◽  
Discrete Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

Witts Theorem for Quadratic Forms Over Non-Dyadic Discrete Valuation Rings

1977 ◽  
Vol 29 (5) ◽  
pp. 928-936
Author(s):  
David Mordecai Cohen
Keyword(s):  
Bilinear Form ◽  
Maximal Ideal ◽  
Quadratic Forms ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Symmetric Bilinear Form ◽  
Finitely Generated ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Group Of Isometries

Let R be a discrete valuation ring, with maximal ideal pR, such that ½ ϵ R. Let L be a finitely generated R-module and B : L × L → R a non-degenerate symmetric bilinear form. The module L is called a quadratic module. For notational convenience we shall write xy = B(x, y). Let O(L) be the group of isometries, i.e. all R-linear isomorphisms φ : L → L such that B((φ(x), (φ(y)) = B(x, y).

On Condensed Noetherian Domains Whose Integral Closures are Discrete Valuation Rings

1989 ◽  
Vol 32 (2) ◽  
pp. 166-168 ◽  
Author(s):  
Christian Gottlieb
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Discrete Valuation Ring ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Integral Closures

AbstractA condensed domain is an integral domain such that IJ = {xy : x ∊ I, y ∊ J } holds for each pair I, J of ideals. We prove that, under suitable conditions, a subring of a discrete valuation ring is condensed if and only if it contains an element of value 2. We also define the concept strongly condensed.

FINITE POSETS AND THEIR REPRESENTATION ALGEBRAS

2010 ◽  
Vol 20 (01) ◽  
pp. 27-38 ◽  
Author(s):  
MURRAY GERSTENHABER ◽  
MARY SCHAPS
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Dimensional Algebra ◽  
Finite Poset ◽  
Finite Dimensional ◽  
Triangular Representation ◽  
Complete Discrete Valuation Ring ◽  
Characteristic 2 ◽  
Finite Posets

A finite poset (P ≼ S) determines a finite dimensional algebra TP over the field 𝔽 of two elements, with an upper triangular representation. We determine the structure of the radical of the representation algebra A of the monoid (TP,·) over a field of characteristic different from 2. We also consider degenerations of A over a complete discrete valuation ring with residue field of characteristic 2.

Discriminant as a product of local discriminants

2017 ◽  
Vol 16 (10) ◽  
pp. 1750198 ◽  
Author(s):  
Anuj Jakhar ◽  
Bablesh Jhorar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Maximal Ideal ◽  
Valuation Ring ◽  
Approximation Theorem ◽  
Discrete Valuation Ring ◽  
Integral Closure ◽  
Weak Approximation ◽  
Separable Extension ◽  
Ideal Formula ◽  
Valuation Rings ◽  
Basic Equality

Let [Formula: see text] be a discrete valuation ring with maximal ideal [Formula: see text] and [Formula: see text] be the integral closure of [Formula: see text] in a finite separable extension [Formula: see text] of [Formula: see text]. For a maximal ideal [Formula: see text] of [Formula: see text], let [Formula: see text] denote respectively the valuation rings of the completions of [Formula: see text] with respect to [Formula: see text]. The discriminant satisfies a basic equality which says that [Formula: see text]. In this paper, we extend the above equality on replacing [Formula: see text] by the valuation ring of a Krull valuation of arbitrary rank and completion by henselization. In the course of proof, we prove a generalization of the well-known weak Approximation Theorem which is of independent interest as well.

scholarly journals On the Structure of Affine Flat Group Schemes Over Discrete Valuation Rings, II

2020 ◽  
Author(s):  
Phùng Hô Hai ◽  
João Pedro dos Santos
Keyword(s):  
Discrete Valuation Ring ◽  
Free Module ◽  
Galois Groups ◽  
Affine Group ◽  
Group Schemes ◽  
Infinite Type ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
D Modules ◽  
Tannakian Categories

Abstract In the first part of this work [ 12], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of $\mathcal{D}$-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of “infinite type,” Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of $\mathcal{D}$-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs.

scholarly journals Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 744
Author(s):  
Andrei Bura ◽  
Qijun He ◽  
Christian Reidys
Keyword(s):  
Discrete Valuation Ring ◽  
Homology Theory ◽  
Simplicial Homology ◽  
Discrete Valuation Rings ◽  
Relative Homology ◽  
Valuation Rings ◽  
Contraction Procedure ◽  
Chain Maps ◽  
Homology Module ◽  
Invariant Factors

An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex X features a new boundary operator and is formulated over a discrete valuation ring, R. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, Hi(X), and weighted homology, Hi,R(X), in two ways: first, via chain maps, and second, via the relative homology. We compute H0,R(X) by means of a recursive contraction procedure on a weighted spanning tree and H1,R(X) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H1,R(X). The homology module H2,R(X) is naturally obtained from H2(X) via chain maps. Furthermore, we show that all weighted homology modules Hi,R(X) are trivial for i>2. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.

Representations of Finite Posets Over Discrete Valuation Rings

2007 ◽  
Vol 35 (10) ◽  
pp. 3128-3144 ◽  
Author(s):  
David M. Arnold ◽  
Daniel Simson
Keyword(s):  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Finite Posets

scholarly journals On three-variable expanders over finite valuation rings

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Le Quang Ham ◽  
Nguyen Van The ◽  
Phuc D. Tran ◽  
Le Anh Vinh
Keyword(s):  
Valuation Ring ◽  
Product Type ◽  
Quadratic Polynomial ◽  
Valuation Rings

AbstractLet {\mathcal{R}} be a finite valuation ring of order {q^{r}}. In this paper, we prove that for any quadratic polynomial {f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form {axy+R(x)+S(y)+T(z)} for some one-variable polynomials {R,S,T}, we have|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}for any {A,B,C\subset\mathcal{R}}. We also study the sum-product type problems over finite valuation ring {\mathcal{R}}. More precisely, we show that for any {A\subset\mathcal{R}} with {|A|\gg q^{r-\frac{1}{3}}} then {\max\{|AA|,|A^{d}+A^{d}|\}}, {\max\{|A+A|,|A^{2}+A^{2}|\}}, {\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}, and {|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f.

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

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