ABSTRACT. The Bohnenblust-Hille inequality says that the ℓ 2m m+1 -norm of the coefficients of an m-homogeneous polynomial P on C n is bounded by P ∞ times a constant independent of n, where · ∞ denotes the supremum norm on the polydisc D n . The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be C m for some C > 1. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc D n behaves asymptotically as (log n)/n modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies log n : n a positive integer ≤ N is √ N exp{(−1/ √ 2 + o(1)) √ log N log log N } as N → ∞.
Abstract. We give a geometric description of the interpolating varieties for the algebra of Fourier transforms of distributions (or Beurling ultradistributions) with compact support on the real line.
In a first (theoretical) part of this paper, we prove a number of constraints on hypothetical counterexamples to the Casas-Alvero conjecture, building on ideas of Graf von Bothmer, Labs, Schicho and van de Woestijne that were recently reinterpreted by Draisma and de Jong in terms of p-adic valuations. In a second (computational) part, we present ideas improving upon Diaz-Toca and Gonzalez-Vega's Gröbner basis approach to the Casas-Alvero conjecture. One application is an extension of the proof of Graf von Bothmer et al. to the cases 5p k , 6p k and 7p k (that is, for each of these cases, we elaborate the finite list of primes p for which their proof is not applicable). Finally, by combining both parts, we settle the Casas-Alvero conjecture in degree 12 (the smallest open case).
ABSTRACT. We use L 2 estimates for the∂ equation to find geometric conditions on discrete interpolating varieties for weighted spaces A p (C) of entire functions such that |f (z)| ≤ Ae Bp(z) for some A, B > 0. In particular, we give a characterization when p(z) = e |z| and more generally when ln p(e r ) is convex and ln p(r) is concave.
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