scholarly journals Correction to: Stability of Valuations: Higher Rational Rank

2019 ◽  
Author(s):  
Chi Li ◽  
Chenyang Xu
Keyword(s):  
Rational Rank

scholarly journals Generalization of real interval matrices to other fields

2019 ◽  
Vol 35 ◽  
pp. 285-296
Author(s):  
Elena Rubei
Keyword(s):  
Full Rank ◽  
Maximal Rank ◽  
Interval Matrix ◽  
Real Interval ◽  
Rational Matrix ◽  
Rank One ◽  
Rational Rank ◽  
Q Matrix ◽  
Rational Interval ◽  
Rank Of A Matrix

An interval matrix is a matrix whose entries are intervals in $\R$. This concept, which has been broadly studied, is generalized to other fields. Precisely, a rational interval matrix is defined to be a matrix whose entries are intervals in $\Q$. It is proved that a (real) interval $p \times q$ matrix with the endpoints of all its entries in $\Q$ contains a rank-one matrix if and only if it contains a rational rank-one matrix, and contains a matrix with rank smaller than $\min\{p,q\}$ if and only if it contains a rational matrix with rank smaller than $\min\{p,q\}$; from these results and from the analogous criterions for (real) inerval matrices, a criterion to see when a rational interval matrix contains a rank-one matrix and a criterion to see when it is full-rank, that is, all the matrices it contains are full-rank, are deduced immediately. Moreover, given a field $K$ and a matrix $\al$ whose entries are subsets of $K$, a criterion to find the maximal rank of a matrix contained in $\al$ is described.

scholarly journals On the index of Dirac operators on arithmetic quotients

1997 ◽  
Vol 56 (3) ◽  
pp. 489-497 ◽  
Author(s):  
Anton Deitmar
Keyword(s):  
Trace Formula ◽  
Special Interest ◽  
Error Term ◽  
Dirac Operators ◽  
Index Formula ◽  
Real Rank ◽  
Present Evidence ◽  
Index Theorems ◽  
Compact Case ◽  
Rational Rank

The aim of this note is to show how the trace formula of Arthur-Selberg can be used to derive index theorems for noncompact arithmetic manifolds. Of special interest is the question, under which circumstances there is an index formula without error term, that is, of the same shape as in the compact case. We shall thus present evidence for the hypothesis that the error term for the Euler operator vanishes in the case that the rational rank is smaller than the real rank.

scholarly journals Semistable reduction for overconvergent F-isocrystals, IV: local semistable reduction at nonmonomial valuations

2011 ◽  
Vol 147 (2) ◽  
pp. 467-523 ◽  
Author(s):  
Kiran S. Kedlaya
Keyword(s):  
Positive Characteristic ◽  
Degree Zero ◽  
Local Problem ◽  
Finite Cover ◽  
Semistable Reduction ◽  
Numerical Invariants ◽  
Rational Rank ◽  
Logarithmic Singularities ◽  
Meromorphic Connection ◽  
Field Of Positive Characteristic

AbstractWe complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree zero. We do this by studying the variation of some numerical invariants attached to p-adic differential modules, analogous to the irregularity of a complex meromorphic connection. This allows for an induction on the transcendence defect of the valuation, i.e., the discrepancy between the dimension of the variety and the rational rank of the valuation.

scholarly journals Generating sequences and semigroups of valuations on 2 dimensional normal local rings

2018 ◽  
Author(s):  
◽  
Arpan Dutta
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Local Rings ◽  
Closed Field ◽  
Quotient Singularity ◽  
Two Dimensional ◽  
Invariant Ring ◽  
Rational Rank ◽  
Generating Sequence ◽  
Generating Sequences

In this thesis we develop a method for constructing generating sequences for valuations dominating the ring of a two dimensional quotient singularity. Suppose that K is an algebraically closed field of characteristic zero, K[X, Y] is a polynomial ring over K and v is a rational rank 1 valuation of the field K(X, Y) which dominates K[X, Y](X,Y) . Given a finite Abelian group H acting diagonally on K[X, Y], and a generating sequence of v in K[X, Y] whose members are eigenfunctions for the action of H, we compute a generating sequence for the invariant ring K[X, Y]H. We use this to compute the semigroup SK[X,Y ]H (v) of values of elements of K[X, Y]H. We further determine when SK[X,Y ]H (v) is a finitely generated SK[X,Y ]H (v)-module.

scholarly journals Normal subgroups in limit groups of prime index

2018 ◽  
Vol 21 (1) ◽  
pp. 83-100 ◽  
Author(s):  
Thomas S. Weigel ◽  
Jhoel S. Gutierrez
Keyword(s):  
Special Kind ◽  
Affirmative Answer ◽  
Limit Group ◽  
Limit Groups ◽  
Number Of Generators ◽  
Analogous Question ◽  
Minimum Number ◽  
Original Question ◽  
Rational Rank ◽  
Torsion Free

AbstractMotivated by their study of pro-plimit groups, D. H. Kochloukova and P. A. Zalesskii formulated in [15, Remark after Theorem 3.3] a question concerning the minimum number of generators{d(N)}of a normal subgroupNof prime indexpin a non-abelian limit groupG(see Question*). It is shown that the analogous question for the rational rank has an affirmative answer (see Theorem A). From this result one may conclude that the original question of Kochloukova and Zalesskii has an affirmative answer if the abelianization{G^{\mathrm{ab}}}ofGis torsion free and{d(G)=d(G^{\mathrm{ab}})}(see Corollary B), or ifGis a special kind of one-relator group (see Theorem D).

VALUATIONS OF k((X1,…,Xn)) WITH PREASSIGNED GROUP OF VALUES

2004 ◽  
Vol 03 (04) ◽  
pp. 453-468 ◽  
Author(s):  
A. GRANJA
Keyword(s):  
Abelian Group ◽  
Power Series ◽  
Algebraic Extension ◽  
Power Series Ring ◽  
Quotient Field ◽  
Series Ring ◽  
Rational Rank

Let Γ be a totally ordered abelian group of finite rational rank r. Consider s and n two non-negative integers with r+s≤n and denote by K a field. The main purpose of this paper is to construct a s-dimensional valuation v of the quotient field K((X1,…,Xn)) of the corresponding power series ring K[[X1,…,Xn]] such that v birrationally dominates K[[X1,…,Xn]] and has Γ as group of values. This is possible except for the case in which Γ=ℤ is the group of the integers, s=0, n≥2 and K does not admit an infinite algebraic extension. In this last case, we show that there does not exist such a valuation.

scholarly journals A rational rank four demand system

2003 ◽  
Vol 18 (2) ◽  
pp. 127-135 ◽  
Author(s):  
Arthur Lewbel
Keyword(s):  
Demand System ◽  
Rational Rank

scholarly journals Truncated Local Uniformization of Formal Integrable Differential Forms

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
F. Cano ◽  
M. Fernández-Duque
Keyword(s):  
Differential Forms ◽  
Newton Polygon ◽  
Codimension One ◽  
Inductive Procedure ◽  
Rank One ◽  
Local Uniformization ◽  
Rational Rank ◽  
Integrable Differential Forms

AbstractWe prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential 1-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.

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