Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.53 (2021)

The journal Rendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics. Rendiconti dell’Istituto di Matematica dell’Universit`a di Trieste has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. In 2008 the Dipartimento di Matematica e Informatica, the owner of the journal, decided to renew it. The name of the journal however remained unchanged, but the subtitle An International Journal of Mathematics was added. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in. The articles published by Rendiconti dell’Istituto di Matematica dell’Universit`a di Trieste are reviewed/indexed by MathSciNet, Zentralblatt Math, Scopus, OpenStarTs.

We introduce a mathematical structure called Euclidean Universe. This structure provides a basic framework for Non- Archimedean Mathematics and Nonstandard Analysis.

If the quaternary quartic equation 9 · (u² + 7v²)² − 7 · (r² + 7s²)² = 2 (*) which M. Davis put forward in 1968 has only finitely many solutions in integers, then — it was observed by M. Davis, Yu. V. Matiyasevich, and J. Robinson in 1976 — every listable set would turn out to admit a single-fold Diophantine representation. In 1995, D. Shanks and S. S. Wagstaff conjectured that (*) has infinitely many solutions; while in doubt, it seemed wise to us to single out new candidates for the role of “rule-them-all equation”. We offer three new quaternary quartic equations, each obtained by much the same recipe which led to (*). The significance of those can be supported by arguments analogous to the ones found in Davis’s original paper; moreover, they might play a key role in settling the conjecture that every listable set has a single-fold (or, at least, a finite-fold) representation. Directly from the unproven assertion that any of the novel equations has only finitely many solutions in integers, one can construct a Diophantine relation of exponential growth, as we show in detail for one, namely 3 · (r² + 3s²)² − (u² + 3v²)² = 2, of the new candidate rule-them-all equations. An account of Julia Robinson’s earliest Diophantine reduction of exponentiation to any relation of exponential growth is also included, for the sake of self-containedness.