Введение
Высота полинома
In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as
where factorizes over the complex numbers as
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 for on the unit circle (i. e., ):
By extension, the Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply
The Mahler measure is named after the German born Australian mathematician Kurt Mahler.
В математике мера Малера полинома с комплексными коэффициентами определяется как
In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as
where factorizes over the complex numbers as
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 for on the unit circle (i. e., ):
By extension, the Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply
The Mahler measure is named after the German born Australian mathematician Kurt Mahler.
где полином разлагается на множители над комплексными числами как
In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as
where factorizes over the complex numbers as
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 for on the unit circle (i. e., ):
By extension, the Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply
The Mahler measure is named after the German born Australian mathematician Kurt Mahler.
Меру Малера можно рассматривать как своего рода функцию высоты. Используя формулу Йенсена, можно доказать, что эта мера также равна геометрическому среднему значений для на единичной окружности (т.е.):
In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as
where factorizes over the complex numbers as
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 for on the unit circle (i. e., ):
By extension, the Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply
The Mahler measure is named after the German born Australian mathematician Kurt Mahler.
По расширению, мера Малера алгебраического числа определяется как мера Малера его минимального полинома над . В частности, если является числом Пизота или числом Салема, то его мера Малера просто равна .
In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as
where factorizes over the complex numbers as
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 for on the unit circle (i. e., ):
By extension, the Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply
The Mahler measure is named after the German born Australian mathematician Kurt Mahler.
Мера Малера названа в честь немецкого математика австралийского происхождения Курта Малера.
In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as
where factorizes over the complex numbers as
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 for on the unit circle (i. e., ):
By extension, the Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply
The Mahler measure is named after the German born Australian mathematician Kurt Mahler.
Предложение Бойда
Бойд привел более общие утверждения, чем вышеуказанная теорема. Он отметил, что классическая теорема Кронекера, характеризующая монические многочлены с целыми коэффициентами, все корни которых лежат внутри единичного диска, может рассматриваться как характеристика тех многочленов от одной переменной, мера которых равна ровно 1, и что этот результат обобщается на многочлены от нескольких переменных. Топологическая энтропия (которая равна энтропии по мере) этого действия, , задается мерой Малера (или бесконечна). В случае циклического модуля для ненулевого многочлена формула, доказанная Линдом, Шмидтом и Уордом, дает , логарифмическую меру Малера. В общем случае энтропия действия выражается как сумма логарифмических мер Малера по генераторам главных связанных простых идеалов модуля. Как ранее отмечал Линд в случае автоморфизма одной компактной группы, это означает, что множество возможных значений энтропии таких действий либо равно всему , либо является счетным множеством, в зависимости от решения проблемы Лемера. Линд также показал, что бесконечномерный тор либо имеет эргодические автоморфизмы с конечной положительной энтропией, либо имеет только автоморфизмы с бесконечной энтропией, в зависимости от решения проблемы Лемера.