Description
Dear Colleague,
It is our pleasure to invite you to the next talk in our Online Number Theory Seminar, on
7 June (Friday) at 17:00 CET (8:00 PDT, 11:00 EDT, 15:00 GMT, 18:00 Israel Daylight Time, 20:30 Indian Standard Time; already Saturday 2:00 AEDT, 4:00 NZDT)
Speaker: N. Hirata-Kohno (Nihon University)
Title: Number of the solutions of $S$-unit equation in two variables
Abstract: This is a joint work with Makoto Kawashima, Anthony Poels and Yukiko Washio.
We apply the explicit Pade approximation constructed for binomial functions by the second and the third authors,
to give a new upper bound for the number of the solutions of the $S$-unit equation, that refines the bound due to J.-H. Evertse.
Let $K$ be a number field of degree $m$ and let $a, b$ be non-zero elements of $K.$ Consider a finite set $S$ of places
of $K$ containing all the Archimedean ones. Denote by $s$ its cardinality and by $U_S$ the set of the $S$-units in $K.$
In 1984, Evertse proved that the $S$-unit equation $ax+by=1\, (x, y \in U_S)$ has at most $3\times 7^{m+2s}$ solutions.
We refine for any positive integers $m, s$ showing that the equation $ax+by=1$ has at most
$(3.1+5 (3.4)^{m}) \times 45^s$ solutions $(x, y) \in U_S^2.$
We use the result proven by Loher and Masser in 2004 to obtain a further improvement:
$(3.1+68 m\log m (1.5)^m)45^s,$ which is smaller than the bound above when $m\geq 6$ and $s\geq 1.$
Meeting link: https://unideb.webex.com/unideb/j.php?MTID=ma6eabcaff228e1d6352526e05d434e4a
Meeting number: 2730 998 6582
Please, share the link with colleagues who are interested, tell us if you have a proposal for a talk, and
also inform us in case you are not willing to receive announcements of these events. The email address of the
seminar is ntrg@science.unideb.hu . We keep the program updated at https://ntrg.math.unideb.hu/seminar.html ,
where (upon the agreement of the Speaker), you can also find the video and the slides of the talk.
We are looking forward to meeting you, with kind regards,
the organizers (K.Győry, Á.Pintér, L.Hajdu, A.Bérczes, Sz.Tengely, I.Pink, Debrecen Number Theory Research Group , University of Debrecen)