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

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

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$is equivalent to a statement about an arbitrary pair of points$\alpha,\beta\in\sper\ A$and their separating ideal$$; we refer to this statement as the Local Pierce-Birkhoff conjecture at$\alpha,\beta$. In this paper, for each pair$(\alpha,\beta)$with$ht()=\dim A$, we define a natural number, called complexity of$(\alpha,\beta)$. Complexity 0 corresponds to the case when one of the points$\alpha,\beta$is monomial; this case was already settled in all dimensions in a preceding paper. Here we introduce a new conjecture, called the Strong Connectedness conjecture, and prove that the strong connectedness conjecture in dimension n-1 implies the connectedness conjecture in dimension n in the case when$ht()$is less than n-1. We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce--Birkhoff conjectures in any dimension in the case when$ht()$less than 2. Finally, we prove the Connectedness (and hence also the Pierce--Birkhoff) conjecture in the case when dimension of A is equal to$ht()=3$, the pair$(\alpha,\beta)$is of complexity 1 and$A$ is excellent with residue field the field of real numbers

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

The thick-thin decomposition and the bilipschitz classification of normal surface singularities

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ 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 $(X,0)$ in terms of its topology and a finite list of numerical bilipschitz invariants.Comment: Minor corrections. To appear in Acta Mathematic

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

Seasonal variation of carbon monoxide in northern Japan: Fourier transform IR measurements and source-labeled model calculations

Tropospheric carbon monoxide (CO) was measured throughout 2001 using groundbased Fourier transform IR (FTIR) spectrometers at Moshiri 44.4N and Rikubetsu 43.5N) observatories in northern Japan, which are separated by 150 km. Seasonal and day-to-day variations of CO are studied using these data, and contributions from various CO sources are evaluated using three-dimensional global chemistry transport model (GEOS-CHEM) calculations. Seasonal maximum and minimum FTIR-derived tropospheric CO amounts occurred in April and September, respectively. The ratio of partial column amounts between the 0–4 and 0–12 km altitude ranges is found to be slightly greater in early spring. The GEOS-CHEM model calculations generally reproduce these observed features. Source-labeled CO model calculations suggest that the observed seasonal variation is caused by seasonal contributions from various sources, in addition to a seasonal change in chemical CO loss by OH. Changes in meteorological fields largely control the relative importance of various source contributions. The contributions from fossil fuel (FF) combustion in Asia and photochemical CO production have the greatest yearly averaged contribution at 1 km among the CO sources (31% each). The Asian FF contribution increases from winter to summer, because weak southwesterly wind in summer brings more Asian pollutants to the observation sites. The seasonal variation from photochemical CO production is small (±17% at 1 km), likely because of concurrent increases (decreases) of photochemical production and loss rates in summer (winter), with the largest contribution between August and December. The contribution from intercontinental transport of European FF combustion CO is found to be comparable to that of Asian FF sources in winter. Northwesterly wind around the Siberian high in this season brings pollutants from Europe directly to Japan, in addition to southward transport of accumulated pollution from higher latitudes. The influences are generally greater at lower altitudes, resulting in a vertical gradient in the CO profile during winter. The model underestimates total CO by 12–14% between March and June. Satellite-derived fire-count data and the relationship between FTIR-derived HCN and CO amounts are generally consistent with biomass burning influences, which could have been underestimated by the model calculations

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