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Seminars

Number Theory Seminar

FOM/FOD problem for cubic rational functions, and multiplicative relations in semigroup orbits

2026/05/26 (Tue.) 10:30~11:30
  • Date : 2026/05/26 (Tue.) 10:30~11:30
  • Location : Seminar Room 617, Institute of Mathematics (NTU Campus)
  • Speaker : Yu Yasufuku (Waseda University)
  • Organizer : Julie Tzu-Yueh Wang (AS)
  • Abstract :
    I discuss two topics in arithmetic dynamics. The first part is about the field-of-definition (FOD) and the field-of-moduli (FOM). Silverman showed that for each rational function $\phi$ there exists a linear fractional transformation $f$ such that $f\circ \phi \circ f^{-1}$ is defined over a quadratic extension of FOM. Moreover, Silverman showed FOD can be made equal to FOM when the degree of $\phi$ is even. We show an explicit criterion for FOD to be equal to the FOM in the case of some cubic functions, and illustrate with some examples.

    The second part, which is a joint work with Jorge Mello (Oakland Univ.), is about multiplicative dependence of orbits under several morphisms on $\mathbb P^n$. We show, in a quantitative way, the sparsity of multiplicative dependence in orbits is implied by the non-density of integral points in orbits. This can be viewed as a semigroup-dynamical and a higher-dimensional version of Bérczes—Ostafe—Shparlinski—Silverman.
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