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LOWER BOUNDS FOR THE MAHLER MEASURE AND INERTIA DEGREES OF PRIMES

Part of: Algebraic number theory: global fields Algebraic number theory: local and $p$-adic fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  02 December 2024

SHANTA LAISHRAM Open the ORCID record for SHANTA LAISHRAM [Opens in a new window]  and
GOREKH PRASAD Open the ORCID record for GOREKH PRASAD [Opens in a new window]
SHANTA LAISHRAM
Affiliation:
Indian Statistical Institute, New Delhi 110016, India e-mail: [email protected]
GOREKH PRASAD*
Affiliation:
Harish-Chandra Research Institute, A CI of Homi Bhaba National Institute, Prayagraj 211019, India
*
e-mail: [email protected]
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Abstract

We investigate the relationship between lower bounds for the Mahler measure and splitting of primes, and prove various lower bounds for the Mahler measure of algebraic integers in terms of the least common multiples of all inertia degrees of primes. The results generalise work of the second author and Kumar [‘Lehmer’s problem and splitting of rational primes in number fields’, Acta Math. Hungar. 169(2) (2023), 349–358].

Keywords

Lehmer’s problemMahler measureabsolute Weil heightprime factorisationinertia degree

MSC classification

Primary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Secondary: 11G50: Heights 11S15: Ramification and extension theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 12
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author acknowledges the support of a SERB CRG Grant while working during the project.

References

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