Ideals Generated by Principal Minors.

Let X be a square matrix of indeterminates. Let K[X] denote the polynomial ring in those indeterminates over an algebraically closed field, K. A minor is principal means it is defined by the same row and column indices. We prove various statements about ideals generated by principal minors of a fixed size t. When t=2 the resulting quotient ring is a normal complete intersection domain. When t>2 we break the problem into cases by intersecting with the locally closed variety of rank r matrices. We show when r=n for any t, there is a K-automorphism of that maps the ideal generated by size t principal minors to the ideal generated by size n-t principal minors, inducing an isomorphism on the respectively defined schemes. When t=r we factor a matrix in the algebraic set as the product of its row space matrix, an invertible size t matrix, and its column space matrix. We show that for the analysis of components it is enough to consider irreducible algebraic sets in the product of two Grassmannians, Grass(t,n). For t=r we also observe the connection between such decompositions and matroid theory.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108987/1/wheeles_1.pd

Ideals Generated by Principal Minors

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$ denote the ideal generated by the size $t$ principal minors of $X$. When $t=2$ the resulting quotient ring $K[X]/\mathfrak P_2$ is a normal complete intersection domain. When $t>2$ we break the problem into cases depending on a fixed rank, $r$, of $X$. We show when $r=n$ for any $t$, the respective images of $\mathfrak P_t$ and $\mathfrak P_{n-t}$ in the localized polynomial ring, where we invert $\det X$, are isomorphic. From that we show the algebraic set given by $\mathfrak P_{n-1}$ has a codimension $n$ component, plus a codimension 4 component defined by the determinantal ideal (which is given by all the submaximal minors of $X$). When $n=4$ the two components are linked, and we prove some consequences

Toric and tropical Bertini theorems in positive characteristic

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field \kk(t_1,\dots,t_d) of rational functions over an algebraically closed field \kk of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the d-connectivity result for tropical varieties from that paper

Prompt Optical Detection of GRB 050401 with ROTSE-IIIa

The ROTSE-IIIa telescope at Siding Spring Observatory, Australia, detected prompt optical emission from Swift GRB 050401. In this letter, we present observations of the early optical afterglow, first detected by the ROTSE-IIIa telescope 33 s after the start of gamma-ray emission, contemporaneous with the brightest peak of this emission. This GRB was neither exceptionally long nor bright. This is the first prompt optical detection of a GRB of typical duration and luminosity. We find that the early afterglow decay does not deviate significantly from the power-law decay observable at later times, and is uncorrelated with the prompt gamma-ray emission. We compare this detection with the other two GRBs with prompt observations, GRB 990123 and GRB 041219a. All three bursts exhibit quite different behavior at early times.Comment: 4 pages, 3 figures. Accepted for publication in ApJ Letter

The Science Case for PILOT I: Summary and Overview

Original article can be found at: http://www.publish.csiro.au/?nid=139&aid=108 DOI: 10.1071/AS08048 [Open access article]PILOT (the Pathfinder for an International Large Optical Telescope) is a proposed 2.5-m optical/infrared telescope to be located at Dome C on the Antarctic plateau. Conditions at Dome C are known to be exceptional for astronomy. The seeing (above ∼30 m height), coherence time, and isoplanatic angle are all twice as good as at typical mid-latitude sites, while the water-vapour column, and the atmosphere and telescope thermal emission are all an order of magnitude better. These conditions enable a unique scientific capability for PILOT, which is addressed in this series of papers. The current paper presents an overview of the optical and instrumentation suite for PILOT and its expected performance, a summary of the key science goals and observational approach for the facility, a discussion of the synergies between the science goals for PILOT and other telescopes, and a discussion of the future of Antarctic astronomy. Paper II and Paper III present details of the science projects divided, respectively, between the distant Universe (i.e. studies of first light, and the assembly and evolution of structure) and the nearby Universe (i.e. studies of Local Group galaxies, the Milky Way, and the Solar System).Peer reviewe

Optical Lightcurve & Cooling Break of GRB 050502A

We present lightcurves of the afterglow of GRB050502A, including very early data at t-t_{GRB} < 60s. The lightcurve is composed of unfiltered ROTSE-IIIb optical observations from 44s to 6h post-burst, R-band MDM observations from 1.6 to 8.4h post-burst, and PAIRITEL J H K_s observations from 0.6 to 2.6h post-burst. The optical lightcurve is fit by a broken power law, where t^{alpha} steepens from alpha = -1.13 +- 0.02 to alpha = -1.44 +- 0.02 at \~5700s. This steepening is consistent with the evolution expected for the passage of the cooling frequency nu_c through the optical band. Even in our earliest observation at 44s post-burst, there is no evidence that the optical flux is brighter than a backward extrapolation of the later power law would suggest. The observed decay indices and spectral index are consistent with either an ISM or a Wind fireball model, but slightly favor the ISM interpretation. The expected spectral index in the ISM interpretation is consistent within 1 sigma with the observed spectral index beta = -0.8 +- 0.1; the Wind interpretation would imply a slightly (~2 sigma) shallower spectral index than observed. A small amount of dust extinction at the source redshift could steepen an intrinsic spectrum sufficiently to account for the observed value of beta. In this picture, the early optical decay, with the peak at or below 4.7e14 Hz at 44s, requires very small electron and magnetic energy partitions from the fireball.Comment: 22 pages, including 3 tables and 1 figure, Accepted by Ap

The Science Case for PILOT II: the Distant Universe

PILOT (the Pathfinder for an International Large Optical Telescope) is a proposed 2.5-m optical/infrared telescope to be located at Dome C on the Antarctic plateau. The atmospheric conditions at Dome C deliver a high sensitivity, high photometric precision, wide-field, high spatial resolution, and high-cadence imaging capability to the PILOT telescope. These capabilities enable a unique scientific potential for PILOT, which is addressed in this series of papers. The current paper presents a series of projects dealing with the distant (redshift >1) Universe, that have been identified as key science drivers for the PILOT facility. The potential for PILOT to detect the first populations of stars to form in the early Universe, via infrared projects searching for pair-instability supernovae and gamma-ray burst afterglows, is investigated. Two projects are proposed to examine the assembly and evolution of structure in the Universe: an infrared survey searching for the first evolved galaxies at high redshift, and an optical survey aimed at characterising moderate-redshift galaxy clusters. Finally, a large-area weak-lensing survey and a program to obtain supernova infrared light-curves are proposed to examine the nature and evolution of dark energy and dark matter