#### Title

Polynomial Estimates, Exponential Curves and Diophantine Approximation

#### Document Type

Article

#### Date

2010

#### Embargo Period

9-28-2010

#### Keywords

Complex variables

#### Language

English

#### Disciplines

Mathematics

#### Description/Abstract

Abstract. Let [alpha] [is an element of] (0, 1) \ [the rationals] and *K* = {(*e*^{z}, *e*^{az}) : |*z*| [less than or equal to] 1} [is a subset of] [the complex numbers]^{2}.If *P* is a polynomial of degree *n* in [the complex numbers]^{2}, normalized by ||*P*||_{K} = 1, we obtain sharp estimates for ||*P*||[delta]^{2} in terms of *n*, where [delta]^{2} is the closed unit bidisk. For most [alpha], we show that sup_{p} ||*P*||[subdelta]^{2}[is less than or equal to] exp (*Cn*^{2}log *n*). However, for [alpha] in subset *S* of the Liouville numbers, sup_{p} ||*P*||[subdelta]^{2} has bigger order of growth. We give a precise characterization of the set *S* and study its properties.

#### Recommended Citation

Coman, Dan and Poletsky, Evgeny A., "Polynomial Estimates, Exponential Curves and Diophantine Approximation" (2010). *Mathematics Faculty Scholarship*. 3.

https://surface.syr.edu/mat/3

#### Source

Metadata from ArXiv.org

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 3.0 License.

## Additional Information

12 pages. To appear in Mathematical Research Letters