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An areal analog of Mahler’s measure. (English) Zbl 1232.11111

Summary: We consider a version of height on polynomial spaces defined by the integral over the normalized area measure on the unit disk. This natural analog of Mahler’s measure arises in connection with extremal problems for Bergman spaces. It inherits many nice properties such as the multiplicative one. However, this height is a lower bound for Mahler’s measure, and it can be substantially lower. We discuss some similarities and differences between the two.

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11C08 Polynomials in number theory
11G50 Heights
30C10 Polynomials and rational functions of one complex variable
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