@prefix psr: . @prefix skos: . @prefix dc: . @prefix xsd: . psr:-VHDD6KJX-8 skos:prefLabel "analytic number theory"@en, "théorie analytique des nombres"@fr ; a skos:Concept ; skos:narrower psr:-FCZQ5314-Q . psr:-SNTKWPJM-D skos:prefLabel "polynôme"@fr, "polynomial"@en ; a skos:Concept ; skos:narrower psr:-FCZQ5314-Q . psr:-FCZQ5314-Q skos:definition """In mathematics, the Mahler measure

{\\\\displaystyle M(p)}

of a polynomial

{\\\\displaystyle p(z)}

with complex coefficients is defined as

{\\\\displaystyle M(p)=|a|\\\\prod _{|\\\\alpha _{i}|\\\\geq 1}|\\\\alpha _{i}|=|a|\\\\prod _{i=1}^{n}\\\\max\\\\{1,|\\\\alpha _{i}|\\\\},}

where

{\\\\displaystyle p(z)}

factorizes over the complex numbers

{\\\\displaystyle \\\\mathbb {C} }

{\\\\displaystyle p(z)=a(z-\\\\alpha _{1})(z-\\\\alpha _{2})\\\\cdots (z-\\\\alpha _{n}).}

The Mahler measure can be viewed as a kind of height function. Using Jensen's formula, it can be proved that this measure is also equal to the geometric mean of

{\\\\displaystyle |p(z)|}

for

{\\\\displaystyle z}

on the unit circle (i.e.,

{\\\\displaystyle |z|=1}

{\\\\displaystyle M(p)=\\\\exp \\\\left(\\\\int _{0}^{1}\\\\ln(|p(e^{2\\\\pi i\\ heta })|)\\\\,d\\ heta \\ ight).}

By extension, the Mahler measure of an algebraic number

{\\\\displaystyle \\\\alpha }

is defined as the Mahler measure of the minimal polynomial of

{\\\\displaystyle \\\\alpha }

over

{\\\\displaystyle \\\\mathbb {Q} }

. In particular, if

{\\\\displaystyle \\\\alpha }

is a Pisot number or a Salem number, then its Mahler measure is simply

{\\\\displaystyle \\\\alpha }

. The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Mahler_measure)"""@en, """En mathématiques, la mesure de Mahler est une mesure de la complexité des polynômes. Elle porte le nom de Kurt Mahler (1903–1988) et était à l'origine utilisée dans la recherche de grands nombres premiers. En raison de la connexion à des valeurs particulières des fonctions L, elle fait l'objet de nombreuses conjectures en théorie analytique des nombres.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/Mesure_de_Mahler)"""@fr ; dc:modified "2024-10-18"^^xsd:date ; skos:broader psr:-SNTKWPJM-D, psr:-VHDD6KJX-8 ; skos:exactMatch , ; skos:prefLabel "mesure de Mahler"@fr, "Mahler measure"@en ; dc:created "2023-08-17"^^xsd:date ; skos:inScheme psr: ; a skos:Concept . psr: a skos:ConceptScheme .