arXiv subjects

William D. Taylor

Publications and source records attributed to William D. Taylor.

12 recordsLinked to original sources

arXiv2026

Arithmetic partial differential operators on ramified extensions of $\ZZ_p$

The notion of $p$-derivation as introduced by Buium has a rich history of applications in arithmetic geometry. Working over $\ZZ_p$, Buium-Ralph-Simanca showed that arithmetic differential operators built from these determine $p$-adic analytic functions and vice versa. Notably, over $\ZZ_p$, there is only one $p$-derivation. In this article, we consider the same question for ramified extensions $A_π$ with uniformizer $π$. This has the effect of passing from ordinary to partial differential operators, since such extensions can enjoy multiple $π$-derivations. We introduce a notion of analytic functions which are naturally determined by these operators in a way analogous to that in the unramified case, but with a significantly richer structure. We show that under mild conditions, analytic functions and partial differential operators determine each other in this setting.

math.NT↗

arXiv2026

New versions of Frobenius and integral closure of ideals

In this article, we define three new operations on ideals which generalize integral closure and Frobenius closure of ideals, whose definitions incorporate an auxiliary ideal and a real parameter. These additional ingredients are common in adjusting old definitions of ideal closures in order to generalize them to pairs, with an eye towards further applications in algebraic geometry. In the case of tight closure, similar generalizations exist due to N. Hara and K.I. Yoshida, as well as A. Vraciu, and in the case of Frobenius closure, to K. Schwede. We study their basic properties and give computationally effective calculations of the adjusted tight, Frobenius, and integral closures in the case of affine semigroup rings in terms of the convex geometry of the associated exponent sets. Finally, as applications, we study submodules of the fraction field of a domain defined in terms of our adjusted closures and an $F$-nilpotent property for pairs.

math.AC↗

arXiv2023

First on-sky results of ERIS at VLT

ERIS (Enhanced Resolution Imager and Spectrograph) is a new adaptive optics instrument installed at the Cassegrain focus of the VLT-UT4 telescope at the Paranal Observatory in Chile. ERIS consists of two near-infrared instruments: SPIFFIER, an integral field unit (IFU) spectrograph covering J to K bands, and NIX, an imager covering J to M bands. ERIS has an adaptive optics system able to work with both LGS and NGS. The Assembly Integration Verification (AIV) phase of ERIS at the Paranal Observatory was carried out starting in December 2021, followed by several commissioning runs in 2022. This contribution will describe the first preliminary results of the on-sky performance of ERIS during its commissioning and the future perspectives based on the preliminary scientific results.

astro-ph.IM↗

arXiv2022

Differentially fixed ideals in toric varieties

This article concerns monomial ideals fixed by differential operators of affine semi-group rings over $\mathbb{C}$. We give a complete characterization of when this happens. Perhaps surprisingly, every monomial ideal is fixed by an infinite set of homogeneous differential operators and is in fact determined by them. This opens up a new tool for studying monomial ideals. We explore applications of this to (mixed) multiplier ideals and other variants as well as give examples of detecting ideal membership in integrally closed powers and symbolic powers of squarefree monomial ideals.

math.AC↗

arXiv2020

High contrast imaging for the enhanced resolution imager and spectrometer (ERIS)

ERIS is a diffraction limited thermal infrared imager and spectrograph for the Very Large Telescope UT4. One of the science cases for ERIS is the detection and characterization of circumstellar structures and exoplanets around bright stars that are typically much fainter than the stellar diffraction halo. Enhanced sensitivity is provided through the combination of (i) suppression of the diffraction halo of the target star using coronagraphs, and (ii) removal of any residual diffraction structure through focal plane wavefront sensing and subsequent active correction. In this paper we present the two coronagraphs used for diffraction suppression and enabling high contrast imaging in ERIS.

astro-ph.IM↗

arXiv2020

Fundamental Results on $s$-Closures

This paper establishes the fundamental properties of the $s$-closures, a recently introduced family of closure operations on ideals of rings of positive characteristic. The behavior of the $s$-closure of homogeneous ideals in graded rings is studied, and criteria are given for when the $s$-closure of an ideal can be described exactly in terms of its tight closure and rational powers. Sufficient conditions are established for the weak $s$-closure to equal to the $s$-closure. A generalization of the Briancon-Skoda theorem is given which compares any two different $s$-closures applied to powers of the same ideal.

math.AC↗

arXiv2017

The s-multiplicity function of 2x2-determinantal rings

This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ a $m \times n$-matrix of variables, we utilize Gröbner bases to give a closed form the length $λ( k[X] / (I_2(X) + \mathfrak{m}^{ \lceil sq \rceil} + \mathfrak{m}^{[q]} ))$ where $s \in \mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $\mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {\it polynomial} function of $q$ for all $s$.

math.AC↗

arXiv2017

On Lower Bounds for $s$-multiplicities

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to the other. The values in this family which do not match these classic variants however are not known yet to be well-behaved. This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings.

math.AC↗

arXiv2017

Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity

We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in toric rings as a certain volume in real space

math.AC↗

arXiv2015

Broad Balmer Wings in BA Hyper/Supergiants Distorted by Diffuse Interstellar Bands: Five Examples in the 30 Doradus Region from the VLT-FLAMES Tarantula Survey

Extremely broad emission wings at H$β$ and H$α$ have been found in VFTS data for five very luminous BA supergiants in or near 30 Doradus in the Large Magellanic Cloud. The profiles of both lines are extremely asymmetrical, which we have found to be caused by very broad diffuse interstellar bands (DIBs) in the longward wing of H$β$ and the shortward wing of H$α$. These DIBs are well known to interstellar but not to many stellar specialists, so that the asymmetries may be mistaken for intrinsic features. The broad emission wings are generally ascribed to electron scattering, although we note difficulties for that interpretation in some objects. Such profiles are known in some Galactic hyper/supergiants and are also seen in both active and quiescent Luminous Blue Variables. No prior or current LBV activity is known in these 30 Dor stars, although a generic relationship to LBVs is not excluded; subject to further observational and theoretical investigation, it is possible that these very luminous supergiants are approaching the LBV stage for the first time. Their locations in the HRD and presumed evolutionary tracks are consistent with that possibility. The available evidence for spectroscopic variations of these objects is reviewed, while recent photometric monitoring does not reveal variability. A search for circumstellar nebulae has been conducted, with an indeterminate result for one of them.

astro-ph.SR↗

arXiv2014

Sher 25: pulsating but apparently alone

The blue supergiant Sher25 is surrounded by an asymmetric, hourglass-shaped circumstellar nebula, which shows similarities to the triple-ring structure seen around SN1987A. From optical spectroscopy over six consecutive nights, we detect periodic radial velocity variations in the stellar spectrum of Sher25 with a peak-to-peak amplitude of ~12 km/s on a timescale of about 6 days, confirming the tentative detec-tion of similar variations by Hendry et al. From consideration of the amplitude and timescale of the signal, coupled with observed line profile variations, we propose that the physical origin of these variations is related to pulsations in the stellar atmosphere, rejecting the previous hypothesis of a massive, short-period binary companion. The radial velocities of two other blue supergiants with similar bipolar nebulae, SBW1 and HD 168625, were also monitored over the course of six nights, but these did not display any significant radial velocity variations.

astro-ph.SR↗

arXiv2011

Two Remarkable Spectroscopic Categories of Young O Stars from the VLT-FLAMES Tarantula Survey

The spectral and spatial characteristics of two special categories of O stars found in the VFTS dataset are presented. One of them comprises very rapid rotators, including several more extreme than any previously known. These objects are distributed around the peripheries of the main 30 Doradus clusters, suggesting a runaway nature for which their radial velocities already provide preliminary supporting evidence. The other category consists of a large number of Vz stars, previously hypothesized on spectroscopic grounds to be on or very near the ZAMS. Their distribution is the inverse of that of the rapid rotators: the Vz are strongly concentrated to the ionizing clusters, plus a newly recognized band of recent and current star formation to the north, which provides strong circumstantial evidence for their extreme youth.

astro-ph.SR↗