### Abstract

Let (Formula presented.) be linearly independent over (Formula presented.), and set (Formula presented.) We prove sharp estimates for the growth of a polynomial of degree n, in terms of (Formula presented.)where (Formula presented.) is the unit polydisk. For all (Formula presented.) with linearly independent entries, we have the lower estimate (Formula presented.)for Diophantine (Formula presented.), we have (Formula presented.)In particular, this estimate holds for almost all (Formula presented.) with respect to Lebesgue measure. The results here generalize those of Coman and Poletsky [6] for (Formula presented.) without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.

Original language | English |
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Journal | Constructive Approximation |

DOIs | |

Publication status | Accepted/In press - Oct 26 2015 |

### Fingerprint

### Keywords

- Bernstein–Walsh inequalities
- Hausdorff dimension
- Liouville vectors
- Polynomial inequalities

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Computational Mathematics

### Cite this

*Constructive Approximation*. https://doi.org/10.1007/s00365-015-9314-2

**Bernstein–Walsh Inequalities in Higher Dimensions over Exponential Curves.** / Kadyrov, Shirali; Lawrence, Mark.

Research output: Contribution to journal › Article

*Constructive Approximation*. https://doi.org/10.1007/s00365-015-9314-2

}

TY - JOUR

T1 - Bernstein–Walsh Inequalities in Higher Dimensions over Exponential Curves

AU - Kadyrov, Shirali

AU - Lawrence, Mark

PY - 2015/10/26

Y1 - 2015/10/26

N2 - Let (Formula presented.) be linearly independent over (Formula presented.), and set (Formula presented.) We prove sharp estimates for the growth of a polynomial of degree n, in terms of (Formula presented.)where (Formula presented.) is the unit polydisk. For all (Formula presented.) with linearly independent entries, we have the lower estimate (Formula presented.)for Diophantine (Formula presented.), we have (Formula presented.)In particular, this estimate holds for almost all (Formula presented.) with respect to Lebesgue measure. The results here generalize those of Coman and Poletsky [6] for (Formula presented.) without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.

AB - Let (Formula presented.) be linearly independent over (Formula presented.), and set (Formula presented.) We prove sharp estimates for the growth of a polynomial of degree n, in terms of (Formula presented.)where (Formula presented.) is the unit polydisk. For all (Formula presented.) with linearly independent entries, we have the lower estimate (Formula presented.)for Diophantine (Formula presented.), we have (Formula presented.)In particular, this estimate holds for almost all (Formula presented.) with respect to Lebesgue measure. The results here generalize those of Coman and Poletsky [6] for (Formula presented.) without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.

KW - Bernstein–Walsh inequalities

KW - Hausdorff dimension

KW - Liouville vectors

KW - Polynomial inequalities

UR - http://www.scopus.com/inward/record.url?scp=84945301611&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945301611&partnerID=8YFLogxK

U2 - 10.1007/s00365-015-9314-2

DO - 10.1007/s00365-015-9314-2

M3 - Article

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

ER -