We study Diophantine equations of type , where both and have at least two distinct critical points (roots of the derivative) and equal critical values at at most two distinct critical points. Various classical families of polynomials are such that satisfies these assumptions for all . Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if has coefficients in a field of characteristic zero, and at least two distinct critical points and all distinct critical values, then the monodromy group of is a doubly transitive permutation group. In particular, cannot be represented as a composition of lower degree polynomials. Several authors have studied monodromy groups of polynomials with some similar properties. We further show that if has at least two distinct critical points and equal critical values at at most two of them, and if with and , then either , or is of special type. In the latter case, in particular, has no three simple critical points, nor five distinct critical points.

Let be two prime integers. In this paper, we investigate the unit groups of the fields and . Furthermore , we give the second 2-class groups of the subextensions of as well as the 2-class groups of the fields and their maximal real subfields.

The arithmetic Kakeya conjecture, formulated by Katz and Tao (Math Res Lett 6(5-6):625-630, 1999), is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in is . In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds.

A key factor in the transmission of infectious diseases is the structure of disease transmitting contacts. In the context of the current COVID-19 pandemic and with some data based on the Hungarian population we develop a theoretical epidemic model (susceptible-infected-removed, SIR) on a multilayer network. The layers include the Hungarian household structure, with population divided into children, adults and elderly, as well as schools and workplaces, some spatial embedding and community transmission due to sharing communal spaces, service and public spaces. We investigate the sensitivity of the model (via the time evolution and final size of the epidemic) to the different contact layers and we map out the relation between peak prevalence and final epidemic size. When compared to the classic compartmental model and for the same final epidemic size, we find that epidemics on multilayer network lead to higher peak prevalence meaning that the risk of overwhelming the health care system is higher. Based on our model we found that keeping cliques/bubbles in school as isolated as possible has a major effect while closing workplaces had a mild effect as long as workplaces are of relatively small size.