Math 2.0

Number theory Definition

Mahler measure

Builds on Complex numbersPolynomial ringRoot of a polynomialFundamental theorem of algebra

Definition

Let f∈ℂ[x]f \in \C[x] be a nonzero polynomial. By the fundamental theorem of algebra, write f=a(x−α1)⋯(x−αn)f = a (x - \alpha_1) \cdots (x - \alpha_n) with a∈ℂ∖{0}a \in \C \setminus \set{0} and roots α1,…,αn∈ℂ\alpha_1, \dots, \alpha_n \in \C. The Mahler measure of ff is M(f)=∣a∣∏i=1nmax⁡(1,∣αi∣).M(f) = \abs{a} \prod_{i=1}^{n} \max(1, \abs{\alpha_i}).

Remarks

By Jensen's formula, M(f)=exp⁡(∫01log⁡∣f(e2πiθ)∣ dθ)M(f) = \exp\left( \int_0^1 \log \abs{f(e^{2 \pi i \theta})} \, d\theta \right). It is multiplicative: M(fg)=M(f)M(g)M(f g) = M(f) M(g). Example: f=2x−1=2(x−12)f = 2x - 1 = 2(x - \frac{1}{2}) gives M(f)=2⋅max⁡(1,12)=2M(f) = 2 \cdot \max(1, \frac{1}{2}) = 2. Non-example of a common confusion: M(f)M(f) is not the largest absolute value of a coefficient; for f=(x+1)2=x2+2x+1f = (x + 1)^2 = x^2 + 2x + 1 that number is 22, while M(f)=1M(f) = 1. If f∈ℤ[x]f \in \Z[x] is nonzero, then M(f)≥1M(f) \ge 1. By a theorem of Kronecker, a nonzero f∈ℤ[x]f \in \Z[x] has M(f)=1M(f) = 1 if and only if ff is ±\pm a product of a power of xx and cyclotomic polynomials.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

Show the path as a graph
Complex numbers, object, no evidence levelComplex numbersobjectGroup, concept, no evidence levelGroupconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionContour integral, definition, no evidence levelContour integraldefinitionPower series in the complex plane, definition, no evidence levelPower series in the c…definitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionRing, definition, no evidence levelRingdefinitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionDisc of convergence, definition, no evidence levelDisc of convergencedefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementPolynomial ring, definition, no evidence levelPolynomial ringdefinitionEstimation lemma, statement, no evidence levelEstimation lemmastatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionSquare root, definition, no evidence levelSquare rootdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionCauchy's integral formula, statement, no evidence levelCauchy's integral for…statementEntire function, definition, no evidence levelEntire functiondefinitionPower series are holomorphic in the disc of convergence, statement, no evidence levelPower series are holo…statementHolomorphic functions are analytic, statement, no evidence levelHolomorphic functions…statementHolomorphic functions are infinitely differentiable, statement, no evidence levelHolomorphic functions…statementCauchy's integral formula for derivatives, statement, no evidence levelCauchy's integral for…statementCauchy's estimates, statement, no evidence levelCauchy's estimatesstatementLiouville's theorem, statement, no evidence levelLiouville's theoremstatementFundamental theorem of algebra, statement, no evidence levelFundamental theorem o…statementMahler measure, definition, no evidence levelMahler measuredefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Group concept
  3. Real numbers object
  4. Set bounded above definition
  5. Complex exponential definition
  6. Contour integral definition
  7. Power series in the complex plane definition
  8. Real and imaginary parts definition
  9. Ring definition
  10. Supremum definition
  11. Circle integral definition
  12. Disc of convergence definition
  13. Completeness axiom statement
  14. Polynomial ring definition
  15. Estimation lemma statement
  16. Existence of square roots of nonnegative reals statement
  17. Polynomial evaluation definition
  18. Square root definition
  19. Root of a polynomial definition
  20. Modulus of a complex number definition
  21. Complex differentiability definition
  22. Holomorphic function definition
  23. Cauchy's integral formula statement
  24. Entire function definition
  25. Power series are holomorphic in the disc of convergence statement
  26. Holomorphic functions are analytic statement
  27. Holomorphic functions are infinitely differentiable statement
  28. Cauchy's integral formula for derivatives statement
  29. Cauchy's estimates statement
  30. Liouville's theorem statement
  31. Fundamental theorem of algebra statement
  32. Mahler measure definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mahler_measure
About this page
Tag1PJS951C8N
URIm2://1PJS951C8N
Kinddefinition
WritingW0 stub
MSC11R06, 11C08
DefinesMahler measure
UsesComplex numbers, Polynomial ring, Root of a polynomial, Fundamental theorem of algebra
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.