Problems around polynomials - the good, the bad and the ugly ...

This is a list of several open problems dealing mainly with univariate polynomials.Comment: 7 pages, no figure

Responses to comments and elaborations of previous posts III

This post is dedicated to the memory of Rabbi Chaim Flom, late rosh yeshiva of Yeshivat Ohr David in Jerusalem. I first met Rabbi Flom thirty years ago when he became my teacher at the Hebrew Youth Academy of Essex County (now known as the Joseph Kushner Hebrew Academy; unfortunately, another one of my teachers from those years also passed away much too young, Rabbi Yaakov Appel). When he first started teaching he was known as Mr. Flom, because he hadn't yet received semikhah (Actually, he had some sort of semikhah but he told me that he didn't think it was adequate to be called "Rabbi" by the students.) He was only at the school a couple of years and then decided to move to Israel to open his yeshiva. I still remember his first parlor meeting which was held at my house. Rabbi Flom was a very special man. Just to give some idea of this, ten years after leaving the United States he was still in touch with many of the students and even attended our weddings. He would always call me when he came to the U.S. and was genuinely interested to hear about my family and what I was working on. He will be greatly missed

Strong magnetic field induces superconductivity in Weyl semi - metal

Microscopic theory of the normal-to-superconductor coexistence line of a 2D two-band Weyl superconductor subjected to magnetic field is constructed. It is shown that a Weyl semi-metal that is nonsuperconducting or having a small critical temperature $T_{c}$ at zero field, might become a superconductor at higher temperature when the magnetic field is tuned to a series of quantized values $H_{n}$. The pairing occurs on Landau levels. It is argued that the phenomenon is much easier detectable in Weyl semi - metals than in parabolic band metals since the quantum limit already has been approaches in several Weyl materials.. An experimental signature of the superconductivity on Landau levels is the reduction of magnetoresistivity. This has already been observed in $Cd_{3}As_{2}$ and several other compounds. The novel kind of quantum oscillations of magnetoresistance detected in $ZrTe_{5}$ is discussed along these lines.Comment: 22 pages, 6 figure

Orbital Magnetism in Two-dimensional Integrable Systems

We study orbital magnetism of a degenerate electron gas in a number of two-dimensional integrable systems, within linear response theory. There are three relevant energy scales: typical level spacing, the energy related to the inverse time of flight across the system, and the Fermi energy. Correspondingly, there are three distinct temperature regimes: microscopic, mesoscopic, and macroscopic. In the first two regimes there are large finite-size effects in the magnetic susceptibility, whereas in the third regime the susceptibility approaches its macroscopic value. In some cases, such as a quasi-one-dimensional strip or a harmonic confining potential, it is possible to obtain analytic expressions for the susceptibility in the entire temperature range.Comment: 28 pages, Latex, 4 Postscript figure

Level Crossing in Random Matrices: I. Random perturbation of a fixed matrix

We consider level crossing in a matrix family $H=H_0+\lambda V$ where $H_0$ is a fixed $N\times N$ matrix and $V$ belongs to one of the standard Gaussian random matrix ensembles. We study the probability distribution of level crossing points in the complex plane of $\lambda$, for which we obtain a number of exact, asymptotic and approximate formulas.Comment: 35 pages, 16 figures; v2: Introduction and sec. 3.3 expanded, refs. adde