Pisot's d-th root's conjecture for function fields and its complex analog

Speaker: 

Julie Tzu-Yueh Wang

Affiliation: 

Academia Sinica

Date: 

Tue, 29/09/2020 - 10:00am

Venue: 

RC-4082, The Red Centre, UNSW

Abstract: 

 Pisot's $d$-th root's conjecture, proved by Zannier in 2000,  can be stated as follows. Let $b$ be a   linear recurrence 
over a number field $k$, and $d\ge2$ be an integer. Suppose that $b(n)$ is the $d$-th power of some element in $k$ for all but finitely many $n$. Then there exists a linear recurrence $a$ over $\overline{k}$ such that $a(n)^{d}=b(n)$ for all $n$.

In this talk,  we propose a function-field analog of this result  and prove it under some "non-triviality'' assumption.  We relate the problem to a  result of Pasten-Wang  on B\"uchi's $d$-th power problem and  develop  a function-field  GCD estimate for multivariable polynomials with ``small coefficients" evaluating at $S$-units arguments.  We will also discuss its complex analog in the notion of  (generalized Ritt's) exponential polynomials.   

This is a joint work with Ji Guo and Chia-Liang Sun.

This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.

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Organisers:
Mike Bennett (University of British Columbia)
Philipp Habegger (University of Basel)
Alina Ostafe (UNSW Sydney)

School Seminar Series: