On Mahler's inequality for the sum of polynomials

Abstract

Suppose that m ≥ 2 is a positive integer. The upper and lower bounds on the best possible constant λ _{m, n} are provided for which the inequality holds for any polynomials f ₁, …, f _m ∈ℂ[z] of degree at most n each. In particular, for n>2m ² and m→∞, it is proved that 2−π²/(12 m ²)+o(m ⁻²)≤λ _{m, n} ≤2−π/(2e m ²)+o(m ⁻²). For m=5 and n ≥ 43, our result yields the bounds 1.967<λ_{5, n} <1.977.

Keywords:

• Mahler's measure
• Complex polynomial
• Gamma function

Acknowledgements

This research was supported in part by the Lithuanian State Studies and Science Foundation and by INTAS grant no. 03-51-5070.

Additional information

Notes on contributors

Artūras Dubickas

Email: [email protected]