A Bombieri-type inequality for Weierstrass sigma functions

Abstract: The Bombieri inequality is a classic inequality in number theory,see [B. Beauzamy, E. Bombieri, P. Enflo, and H. L. Montgomery. Products of polynomials in many variables. Journal of Number Theory, 36(2):219 – 245, 1990)]. The original statement says that given two homogeneous polynomials on $N$ variables $P,Q$ respectively of degree $m$ and $n$, then $$ {\frac {m!n!}{(m+n)!}}\|P\|^{2}\,\|Q\|^{2}\leq \|P\cdot Q\|^{2}\leq \|P\|^{2}\,\|Q\|^{2}, $$ where the norm is the Bombieri-Weyl norm. This inequality admits a rewriting in terms of integrals on the sphere, a property exploited in [U. Etayo. A sharp bombieri inequality, logarithmic energy and well con- ditioned polynomials, 2019]. In a joint work with Joaquim Ortega-Cerd\`a and Haakan Hedenmalm, we use this new definition to generalize the inequality to other Riemannian manifolds, in particular the torus $\mathbb{C}/\Lambda$

classical analysis and ODEsfunctional analysisnumber theory

Audience: researchers in the topic

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