AWe are interested in how small a root of multiplicity k can be for a power series of the form f(z) B 1j _ n=" a i z i with coefficients a i in [k1, 1]. Let r(k) denote the size of the smallest root of multiplicity k possible for such a power series. We show thatWe describe the form that the extremal power series must take and develop an algorithm that lets us compute the optimal root (which proves to be an algebraic number). The computations, for k 27, suggest that the upper bound is close to optimal and that r(k) " 1kc\(kj1), where c l 1n230….

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For any additive character and multiplicative character on a finite field F q , and rational functions f, g in F q (x), we show that the elementary Stepanov-Schmidt method can be used to obtain the corresponding Weil bound for the sum x∈F q \S (g(x)) (f (x)) where S is the set of the poles of f and g. We also determine precisely the number of characteristic values i of modulus q 1/2 and the number of modulus 1.

Abstract. We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let Mn(Z) denote the monic polynomials of degree n with integer coefficients. A monic integer Chebyshev polynomial Mn ∈ Mn(Z) satisfiesand the monic integer Chebyshev constant is then defined byE . This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied.We compute t M (E) for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases. an interval whose endpoints are consecutive Farey fractions. This is characterized byThis should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater.

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