منابع مشابه
On the Maximum Modulus of Polynomials. Ii
Let f(z) := ∑n ν=0 aνz ν be a polynomial of degree n having no zeros in the open unit disc, and suppose that max|z|=1 |f(z)| = 1. How small can max|z|=ρ |f(z)| be for any ρ ∈ [0 , 1)? This problem was considered and solved by Rivlin [4]. There are reasons to consider the same problem under the additional assumption that f ′(0) = 0. This was initiated by Govil [2] and followed up by the present ...
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Introduction 1. THIS paper embodies mainly the proof of results announced previously (Hayman (5)). Let f(z) be a non-constant integral function and p u t
متن کاملWeighted Maximum over Minimum Modulus of Polynomials, Applied to Ray Sequences of Padé Approximants
Let a 0; " > 0. We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set ( r 2 [0; 1] : r maxjtj=1 jP (t)j minjtj=r jP (t)j " ) : Here P is an arbitrary polynomial of degree n. We then apply this to diagonal and ray Padé sequences for functions analytic (or meromorphic) in the unit ball. For example, we show that the diagonal f[n=n]gn=1 sequence provides ...
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Given an undirected graph G, let PG(z) be the polynomial PG(z) = ∑ n (−1)ncnz , where cn is the number of cliques of size n in G. We show that, for every G, the polynomial PG(z) has only one root of smallest modulus. Clique polynomials are related to trace monoids. Indeed, 1 PG(z) is the generating function of the sequence {tn}, where tn is the number of traces of size n in the trace monoid def...
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An open problem in complexity theory is to find the minimal degree of a polynomial representing the n-bit OR function modulo composite m. This problem is related to understanding the power of circuits with MODm gates where m is composite. The OR function is of particular interest because it is the simplest function not amenable to bounds from communication complexity. Tardos and Barrington [TB9...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1974
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1974-0355015-3