A formula for Jack polynomials of the second order

Francisco J. Caro-Lopera; José A. Díaz-García; Graciela González-Farías

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Let χ(m,n) be the unconditional basis constant of the monomial basis $z^{α}$ , α ∈ ℕ₀ⁿ with |α| = m, of the Banach space of all m-homogeneous polynomials in n complex variables, endowed with the supremum norm on the n-dimensional unit polydisc ⁿ. We prove that the quotient of $s u p_{m} \sqrt[m]{s u p_{m} χ (m, n)}$ and √(n/log n) tends to 1 as n → ∞. This reflects a quite precise dependence of χ(m,n) on the degree m of the polynomials and their number n of variables. Moreover, we give an analogous formula for m-linear forms, a...

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Nonreciprocal algebraic numbers of small Mahler's measure

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We prove that there exist at least cd⁵ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most d whose Mahler measures are smaller than 2, where c is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials $1 + x^{r ₁} + \dots + x^{r ₅}$ , where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including $2 r_{j} < r_{j + 1}$ for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct...