176 research outputs found
Key polynomials for simple extensions of valued fields
Let be a simple transcendental extension
of valued fields, where is equipped with a valuation of rank 1. That
is, we assume given a rank 1 valuation of and its extension to
. Let denote the valuation ring of . 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 a countable well ordered set the are called {\bf key
polynomials}. Key polynomials which have no immediate predecessor are
called {\bf limit key polynomials}. Let .
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 then the set of key polynomials has
order type at most , while in the case
this order type is bounded above by , where stands
for the first infinite ordinal.Comment: arXiv admin note: substantial text overlap with arXiv:math/060519
The analogue of Izumi's Theorem for Abhyankar valuations
16 pagesA well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring are linearly comparable to each other. In the present paper we generalize this theorem to the case of Abhyankar valuations with archimedian value semigroup. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. Then we show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the Krull topology and the topology defined by the symbolic powers of an arbitrary ideal
Approximate roots of a valuation and the Pierce-Birkhoff Conjecture
This paper is a step in our program for proving the Piece-Birkhoff Conjecture
for regular rings of any dimension (this would contain, in particular, the
classical Pierce-Birkhoff conjecture which deals with polynomial rings over a
real closed field). We first recall the Connectedness and the Definable
Connectedness conjectures, both of which imply the Pierce - Birkhoff
conjecture. Then we introduce the notion of a system of approximate roots of a
valuation v on a ring A (that is, a collection Q of elements of A such that
every v-ideal is generated by products of elements of Q). We use approximate
roots to give explicit formulae for sets in the real spectrum of A which we
strongly believe to satisfy the conclusion of the Definable Connectedness
conjecture. We prove this claim in the special case of dimension 2. This proves
the Pierce-Birkhoff conjecture for arbitrary regular 2-dimensional rings
Eastern Asian emissions of anthropogenic halocarbons deduced from aircraft concentration data
The Montreal Protocol restricts production of ozone-depleting halocarbons worldwide. Enforcement of the protocol has relied mainly on annual government statistics of production and consumption of these compounds (bottom-up approach). We show here that aircraft observations of halocarbon:CO enhancement ratios on regional to continental scales can be used to infer halocarbon emissions, providing independent verification of the bottom-up approach. We apply this top-down approach to aircraft observations of Asian outflow from the TRACE-P mission over the western Pacific (March April 2001) and derive emissions from eastern Asia (China, Japan, and Korea). We derive an eastern Asian carbon tetrachloride (CCl ) source of 21.5 Gg yr , several-fold larger than previous estimates and amounting to 30% of the global budget for this gas. Our emission estimate for CFC-11 from eastern Asia is 50% higher than inventories derived from manufacturing records. Our emission estimates for methyl chloroform (CH ) and CFC-12 are in agreement with existing inventories. For halon 1211 we find only a strong local source originating from the Shanghai area. Our emission estimates for the above gases result in a 40% increase in the ozone depletion potential (ODP) of Asian emissions relative to previous estimates, corresponding to a 10% global increase in ODP
On the Pierce-Birkhoff Conjecture
International audienceThis paper represents a step in our program towards the proof of the Pierce--Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce-Birkhoff conjecture for a ring A\alpha,\beta\in\sper\ A\alpha,\beta(\alpha,\beta)ht()=\dim A(\alpha,\beta)\alpha,\betaht()ht()ht()=3(\alpha,\beta)A$ is excellent with residue field the field of real numbers
The thick-thin decomposition and the bilipschitz classification of normal surface singularities
We describe a natural decomposition of a normal complex surface singularity
into its "thick" and "thin" parts. The former is essentially metrically
conical, while the latter shrinks rapidly in thickness as it approaches the
origin. The thin part is empty if and only if the singularity is metrically
conical; the link of the singularity is then Seifert fibered. In general the
thin part will not be empty, in which case it always carries essential
topology. Our decomposition has some analogy with the Margulis thick-thin
decomposition for a negatively curved manifold. However, the geometric behavior
is very different; for example, often most of the topology of a normal surface
singularity is concentrated in the thin parts.
By refining the thick-thin decomposition, we then give a complete description
of the intrinsic bilipschitz geometry of in terms of its topology and a
finite list of numerical bilipschitz invariants.Comment: Minor corrections. To appear in Acta Mathematic
Periods and Feynman integrals
We consider multi-loop integrals in dimensional regularisation and the
corresponding Laurent series. We study the integral in the Euclidean region and
where all ratios of invariants and masses have rational values. We prove that
in this case all coefficients of the Laurent series are periods.Comment: 22 pages, appendix added, version to be publishe
A proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing infinite fields
Let R be a regular local ring, containing an infinite field. Let G be a
reductive group scheme over R. We prove that a principal G-bundle over R is
trivial, if it is trivial over the fraction field of R.Comment: Section "Formal loops and affine Grassmannians" is removed as this is
now covered in arXiv:1308.3078. Exposition is improved and slightly
restructured. Some minor correction
Investigation of chlorine radical chemistry in the Eyjafjallajkull volcanic plume using observed depletions in non-methane hydrocarbons
As part of the effort to understand volcanic plume composition and chemistry during the eruption of the Icelandic volcano Eyjafjallajkull, the CARIBIC atmospheric observatory was deployed for three special science flights aboard a Lufthansa passenger aircraft. Measurements made during these flights included the collection of whole air samples, which were analyzed for non-methane hydrocarbons (NMHCs). Hydrocarbon concentrations in plume samples were found to be reduced to levels below background, with relative depletions characteristic of reaction with chlorine radicals (Cl). Recent observations of halogen oxides in volcanic plumes provide evidence for halogen radical chemistry, but quantitative data for free halogen radical concentrations in volcanic plumes were absent. Here we present the first observation-based calculations of Cl radical concentrations in volcanic plumes, estimated from observed NMHC depletions. Inferred Cl concentrations were between 1.3 × 10 and 6.6 × 10 Cl cm. The relationship between NMHC variability and local lifetimes was used to investigate the ratio between OH and Cl within the plume, with [OH]/[Cl] estimated to be ∼37. Copyright 2011 by the American Geophysical Union
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