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On Mahler measures of a self-inversive polynomial and its derivative

On Mahler measures of a self-inversive polynomial and its derivative Let M ( f ) be the Mahler measure of a polynomial f ∈ ℂ ( z ). An old inequality (which is due to Mahler himself) asserts that M ( f ′) ≤ dM ( f ) for each f ∈ ℂ ( z ) of degree d . In contrast, we prove that if f ∈ ℂ ( z ) is a self-inversive polynomial of degree d ≥ 2, then M ( f ′)>( d /2) M ( f ). We also show that this inequality is best possible for d even, namely, that the quotient M ( f ′)/ M ( f ) takes every value in the interval ( d /2, d ) as f runs through reciprocal polynomials f ∈ ℝ ( z ) of degree d . It seems likely that for d odd the constant d /2 is not optimal. For instance, for d = 3, the optimal value of the constant is 1.93867997… instead of 3/2. For each odd d ≥ 5, we prove that there exists a monic reciprocal polynomial f ∈ ℤ ( z ) of degree d such that M ( f ′)<(( d +1)/2) M ( f ). A corresponding problem for L p -norms of a self-inversive polynomial and its derivative is also considered. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

On Mahler measures of a self-inversive polynomial and its derivative

On Mahler measures of a self-inversive polynomial and its derivative

Bulletin of the London Mathematical Society , Volume 42 (2) – Apr 1, 2010

Abstract

Let M ( f ) be the Mahler measure of a polynomial f ∈ ℂ ( z ). An old inequality (which is due to Mahler himself) asserts that M ( f ′) ≤ dM ( f ) for each f ∈ ℂ ( z ) of degree d . In contrast, we prove that if f ∈ ℂ ( z ) is a self-inversive polynomial of degree d ≥ 2, then M ( f ′)>( d /2) M ( f ). We also show that this inequality is best possible for d even, namely, that the quotient M ( f ′)/ M ( f ) takes every value in the interval ( d /2, d ) as f runs through reciprocal polynomials f ∈ ℝ ( z ) of degree d . It seems likely that for d odd the constant d /2 is not optimal. For instance, for d = 3, the optimal value of the constant is 1.93867997… instead of 3/2. For each odd d ≥ 5, we prove that there exists a monic reciprocal polynomial f ∈ ℤ ( z ) of degree d such that M ( f ′)<(( d +1)/2) M ( f ). A corresponding problem for L p -norms of a self-inversive polynomial and its derivative is also considered.

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References (27)

Publisher
Oxford University Press
Copyright
© 2010 London Mathematical Society
Subject
PAPERS
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdp104
Publisher site
See Article on Publisher Site

Abstract

Let M ( f ) be the Mahler measure of a polynomial f ∈ ℂ ( z ). An old inequality (which is due to Mahler himself) asserts that M ( f ′) ≤ dM ( f ) for each f ∈ ℂ ( z ) of degree d . In contrast, we prove that if f ∈ ℂ ( z ) is a self-inversive polynomial of degree d ≥ 2, then M ( f ′)>( d /2) M ( f ). We also show that this inequality is best possible for d even, namely, that the quotient M ( f ′)/ M ( f ) takes every value in the interval ( d /2, d ) as f runs through reciprocal polynomials f ∈ ℝ ( z ) of degree d . It seems likely that for d odd the constant d /2 is not optimal. For instance, for d = 3, the optimal value of the constant is 1.93867997… instead of 3/2. For each odd d ≥ 5, we prove that there exists a monic reciprocal polynomial f ∈ ℤ ( z ) of degree d such that M ( f ′)<(( d +1)/2) M ( f ). A corresponding problem for L p -norms of a self-inversive polynomial and its derivative is also considered.

Journal

Bulletin of the London Mathematical SocietyOxford University Press

Published: Apr 1, 2010

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