Lower bounds for polynomials with algebraic coefficients
نویسندگان
چکیده
منابع مشابه
Lower bounds on the coefficients of Ehrhart polynomials
We present lower bounds for the coefficients of Ehrhart polynomials of convex lattice polytopes in terms of their volume. We also introduce two formulas for calculating the Ehrhart series of a kind of a ”weak” free sum of two lattice polytopes and of integral dilates of a polytope. As an application of these formulas we show that Hibi’s lower bound on the coefficients of the Ehrhart series is n...
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For an integral convex polytope P ⊂ R, we recall i(P, n) = |nP∩Z|the Ehrhart polynomial of P. Let for r = 0, . . . , d,gr(P) be the r-th coefficients ofi(P, n). Martin Henk and Makoto Tagami gave the lower bounds on the coefficientsgr(P) in terms of the volume of P. In general, these bounds are not best possible.However, it is known that in the cases r ∈ {1, 2, d − 2}, these...
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The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form a0 (n−1 0 )1/2 + a1 (n−1 1 )1/2 x + a2 (n−1 2 )1/2 x2 + · · · + an−1 (n−1 n−1 )1/2 xn−1 where aj , j = 0, 1, 2, . . . , n − 1, are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given...
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It is well known that the eigenvalues of any unitary matrix lie on the unit circle. The purpose of this paper is to prove that the eigenvalues of any matrix polynomial, with unitary coefficients, lie inside the annulus A 1 2 ,2 (0) := { z ∈ C | 1 2 < |z| < 2 } . The foundations of this result rely on an operator version of Rouché’s theorem and the intermediate value theorem.
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We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. We apply our technique to bound the loc...
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ژورنال
عنوان ژورنال: Theoretical Computer Science
سال: 1980
ISSN: 0304-3975
DOI: 10.1016/0304-3975(80)90019-5