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Author: damek

constants/40a.md

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 ## Description of constant
 
 Let
+
 $$
 f(x)=\sum_{i=0}^n a_i x^i \;=\; a_n\prod_{i=1}^n (x-\alpha_i)
 $$
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 be a polynomial with complex coefficients. The **Mahler measure** of $f$ is
+
 $$
 M(f)\;:=\;|a_n|\prod_{i=1}^n \max\{1,|\alpha_i|\}.
 $$
+
 <a href="#BDM2007-def-M">[BDM2007-def-M]</a>
 
 For an integer polynomial $f(x)\in\mathbb{Z}[x]$, **Kronecker’s theorem** characterizes the case $M(f)=1$:
+
 $$
 M(f)=1 \quad\Longleftrightarrow\quad f(x)\text{ is a product of cyclotomic polynomials and }x.
 $$
+
 <a href="#BDM2007-kronecker">[BDM2007-kronecker]</a>
 
 Motivated by Lehmer’s question, define **Lehmer’s Mahler measure constant** $C_{40a}$ to be the infimum of Mahler measures strictly larger than $1$ among integer polynomials, and denote it by $L$:
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 $$
 L \;:=\; \inf\bigl\{ M(f)\;:\; f\in\mathbb{Z}[x],\ 1<M(f)\bigr\}.
 $$
+
 <a href="#BDM2007-lehmer-question">[BDM2007-lehmer-question]</a>
 
 Lehmer’s original question (1933) asks whether, for every $\epsilon>0$, there exists an integer polynomial $f$ with
+
 $$
 1<M(f)<1+\epsilon,
 $$
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 which is equivalent to asking whether $L=1$.
 <a href="#BDM2007-lehmer-question">[BDM2007-lehmer-question]</a>