| Speaker: | Yasuaki Gyoda (Nagoya University) |
|---|---|
| Title: | Generalized discrete Markov spectra |
| Date (JST): | Thu, Feb 05, 2026, 15:30 - 17:00 |
| Place: | Seminar Room B |
| Abstract: |
The Markov spectrum is the set of all lower bounds obtained by considering the distance between the integer lattice and a pair of lines through the origin (equivalently, an indefinite binary quadratic form), after an appropriate normalization of this distance. Markov's theorem from 1880 shows that the part of this spectrum lying below $3$ is discrete, and that these lower bounds are constructed in a systematic way from positive integer solutions of the Markov equation$x^2 + y^2 + z^2 = 3xyz$. This result reveals that a geometric problem concerning distances between lines and lattice points is controlled by a arithmetic structure. In this talk, we extend the classical Markov theory using the generalized Markov equation introduced by Gyoda--Matsushita, $x^2 + y^2 + z^2 + k_1 yz + k_2 zx + k_3 xy= (3 + k_1 + k_2 + k_3)\, xyz.$ We show that the numbers appearing in solutions of this equation control the lower bounds of the normalized distances associated with the corresponding indefinite binary quadratic forms, and that a new discrete class arises which contains, as a special case, the classical discrete part below $3$. |
