The Sidon constant for homogeneous polynomials

TL;DR

This paper estimates the Sidon constant for nonzero m-homogeneous polynomials in n variables and applies this to derive a lower bound for the Bohr radius of the polydisc, revealing asymptotic behavior.

Contribution

The paper provides an order-of-magnitude estimate for the Sidon constant for homogeneous polynomials, advancing understanding of polynomial coefficient norms and their supremum norms.

Findings

Estimated the Sidon constant with exponential dependence on m

Derived a lower bound for the Bohr radius in terms of n and log n

Connected polynomial norm estimates to geometric properties of the polydisc

Abstract

The Sidon constant for the index set of nonzero m-homogeneous polynomials P in n complex variables is the supremum of the ratio between the l^1 norm of the coefficients of P and the supremum norm of P in D^n. We present an estimate which gives the right order of magnitude for this constant, modulo a factor depending exponentially on m. We use this result to show that the Bohr radius for the polydisc D^n is bounded from below by a constant times sqrt((log n)/n).