Key Polynomials

The notion of key polynomials was first introduced in 1936 by S. Maclane in the case of discrete rank 1 valuations. . Let K -> L be a field extension and {\nu} a valuation of K. The original motivation for introducing key polynomials was the problem of describing all the extensions {\mu} of {\nu} to L. Take a valuation {\mu} of L extending the valuation {\nu}. In the case when {\nu} is discrete of rank 1 and L is a simple algebraic extension of K Maclane introduced the notions of key polynomials for {\mu} and augmented valuations and proved that {\mu} is obtained as a limit of a family of augmented valuations on the polynomial ring K[x]. In a series of papers, M. Vaqui\'e generalized MacLane's notion of key polynomials to the case of arbitrary valuations {\nu} (that is, valuations which are not necessarily discrete of rank 1). In the paper Valuations in algebraic field extensions, published in the Journal of Algebra in 2007, F.J. Herrera Govantes, M.A. Olalla Acosta and M. Spivakovsky develop their own notion of key polynomials for extensions (K, {\nu}) -> (L, {\mu}) of valued fields, where {\nu} is of archimedian rank 1 (not necessarily discrete) and give an explicit description of the limit key polynomials. Our purpose in this paper is to clarify the relationship between the two notions of key polynomials already developed by vaqui\'e and by F.J. Herrera Govantes, M.A. Olalla Acosta and M. Spivakovsky.Comment: arXiv admin note: text overlap with arXiv:math/0605193 by different author

Auditors perceptions towards the effectiveness of the international standard on auditing 240 Red Flags : evidence from Lebanon

Most of the studies on auditors’ perceptions towards the effectiveness of the international standard on auditing 240 Red Flags (RF) were conducted in developed economies. This research, therefore, fills the gap by aiming to determine whether RF can be helpful for Lebanon certified public accountant (LCPA) working in auditing firms by detecting fraudulent financial reporting (FFR). Data were collected using a questionnaire that was distributed to a random sample of 130 LCPA. The results support that there is a positive significant association between pressures and FFR occurrence in Lebanon. However, there were no support for opportunities and attitudes to be associated with FFR occurrence. Moreover, the findings provide a strong evidence that ISA 240 RF for FFR can help external auditors in detecting material mis-statement (MM) due to fraud in Lebanon. Therefore, the current research recommends LCPA working in the audit firms in Lebanon to focus their efforts more on high quality RF, which will in turn facilitate fraud detection in the financial statements.peer-reviewe

Key polynomials for simple extensions of valued fields

Let $\iota:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with a valuation $\nu$ of rank 1. That is, we assume given a rank 1 valuation $\nu$ of $K$ and its extension $\nu'$ to $L$. Let $(R_\nu,M_\nu,k_\nu)$ denote the valuation ring of $\nu$. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui\'e, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Namely, we associate to $\iota$ a countable well ordered set $$\mathbf{Q}=\{Q_i\}_{i\in\Lambda}\subset K[x];$$ the $Q_i$ are called {\bf key polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are called {\bf limit key polynomials}. Let $\beta_i=\nu'(Q_i)$. We give an explicit description of the limit key polynomials (which may be viewed as a generalization of the Artin--Schreier polynomials). We also give an upper bound on the order type of the set of key polynomials. Namely, we show that if $\operatorname{char}\ k_\nu=0$ then the set of key polynomials has order type at most $\omega$, while in the case $\operatorname{char}\ k_\nu=p>0$ this order type is bounded above by $\omega\times\omega$, where $\omega$ stands for the first infinite ordinal.Comment: arXiv admin note: substantial text overlap with arXiv:math/060519

Ligand enhanced upconversion of near-infrared photons with nanocrystal light absorbers.

We designed and synthesized a tetracene derivative 4-(tetracen-5-yl)benzoic acid (CPT) as a transmitter ligand used in PbS/PbSe nanocrystal (NC) sensitized upconversion of near infrared (NIR) photons. Under optimal conditions, comparing CPT functionalized NCs with unfunctionalized NCs as sensitizers, the upconversion quantum yield (QY) was enhanced 81 times for 2.9 nm PbS NCs from 0.021% to 1.7%, and 11 times for 2.5 nm PbSe NCs from 0.20% to 2.1%. The surface density of CPT controls the solubility of functionalized NCs and the upconversion QY. By increasing the concentration of CPT in the ligand exchange solution, the number of CPT ligand per NC increases. The upconversion QY is maximized at a transmitter density of 1.2 nm-2 for 2.9 nm PbS, and 0.32 nm-2 for 2.5 nm PbSe. Additional transmitter ligands inhibit photon upconversion due to triplet-triplet annihilation (TTA) between two neighboring CPT molecules on the NC surface. 2.1% is the highest reported QY for TTA-based photon upconversion in the NIR with the use of earth-abundant materials

Defect of an extension, key polynomials and local uniformization

International audienceFor all simple and finite extension of a valued field, we prove that its defect is the product of the effective degrees of the complete set of key polynomials associated. As a consequence, we obtain a local uniformization theorem for valuations of rank 1 centered on an equicharacteristic quasi-excellent local domain satisfying some inductive assumptions of lack of defect