Scientists / Open questions

Enrico Bombieri

Number theory; analysis; minimal surfaces

1974 Fields Medal

7 questions

Torsion-anomalous intersections in algebraic tori: the openness and finiteness conjectures of Bombieri, Masser, and Zannier
The Riemann Hypothesis in Li's criterion form: non-negativity of the Li coefficients for every $n\ge 1$
Effective irrationality measures for algebraic numbers: the unresolved questions of Bombieri's 1982 Thue-Siegel-Dyson method, toward an effective Roth's theorem
The analytic complexity of zeta and $L$ -functions after Bombieri: zero-density inputs for the counting-zero method, the $\mathrm{GL}(n)$ ( $n\ge 3$ ) extension, and the exact two-piece Dirichlet polynomial approximation constant
Mechanistic origin of the thickness-limited electro-active fraction and irreversible expansion in polypyrrole(DBS) actuator films
The number of squares and higher powers in an arithmetic progression: Rudin's conjecture and the Bombieri–Granville–Pintz $k$ -th power bounds
Exact determination of the Bombieri numbers $\sigma_{mn}$ : the limit inferior of $(n-\mathrm{Re}\,a_n)/(m-\mathrm{Re}\,a_m)$ at the Koebe function

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