A closer look to Bombieri inequality

14.04.2021 14:00 - 15:30

Ujué Etayo (Universidad de Cantabria)

Bombieri inequality [1] provide us with an answer to the following problem. Let P1 , . . . , Pm ∈ C[x1 , . . . , xn ] be a set of homogeneous polynomials and let ||·|| be the Bombieri-Weyl norm. Define a constant A depending only on the degrees of the polynomials P1 , . . . , Pm and such that (1) ||P1 ||. . .||Pm|| ≤ A||P1 . . . Pm||. The answer given in [1] is proved to be sharp for m = 2. In the recent paper [2] we explore the connections between equation (1) for univariate polynomials with complex coefficients and a set of evenly distributed spherical points, more concretely, minimizers of the discrete logarithmic energy on the sphere S 2 . On a joint work with Håkan Hedenmalm and Joaquim Ortega- Cerdà [3] we rephrase the inequality proposed in [2] on an integral form on the sphere and propose a new family of inequalities inspired on equation (1) that can be stated for any compact Riemannian manifold. Throughout this talk, we will present the different results found in articles [1–3], briefly commenting on some of their proofs and making special emphasis on the geometric intuition behind them.


[1] B. Beauzamy, E. Bombieri, P. Enflo, and H. L. Montgomery, Products of polynomials in many variables, Journal of Number Theory 36 (1990), no. 2, 219 –245.

[2] U. Etayo, A sharp Bombieri inequality, logarithmic energy and well conditioned polynomials, Transactions of the American Mathematical Society (accepted). (2021).

[3] H. Hedenmalm U. Etayo and J. Ortega-Cerdà, Work in progress (2021).


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