##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Annie Carter

#### UCSD

## Two-variable polynomials with dynamical Mahler measure zero

##### Abstract:

Introduced by Lehmer in 1933, the classical Mahler measure of a complex rational function $P$ in one or more variables is given by integrating $\log|P(x_1, \ldots, x_n)|$ over the unit torus. Lehmer asked whether the Mahler measures of integer polynomials, when nonzero, must be bounded away from zero, a question that remains open to this day. In this talk we generalize Mahler measure by associating it with a discrete dynamical system $f: \mathbb{C} \to \mathbb{C}$, replacing the unit torus by the $n$-fold Cartesian product of the Julia set of $f$ and integrating with respect to the equilibrium measure on the Julia set. We then characterize those two-variable integer polynomials with dynamical Mahler measure zero, conditional on a dynamical version of Lehmer's conjecture.

### March 3, 2022

### 9:00 AM

Zoom *only*; see

https://www.math.ucsd.edu/~nts

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