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Multiple Roots of [−1, 1] Power Series
Author(s) -
Beaucoup Frank,
Borwein Peter,
Boyd David W.,
Pinner Christopher
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005857
Subject(s) - multiplicity (mathematics) , mathematics , series (stratigraphy) , power series , combinatorics , root (linguistics) , algebraic number , computation , discrete mathematics , algorithm , geometry , mathematical analysis , paleontology , linguistics , philosophy , biology
We are interested in how small a root of multiplicity k can be for a power series of the form f ( z ) : = 1 + ∑ n = 1 ∞a i z iwith coefficients a i in [−1, 1]. Let r ( k ) denote the size of the smallest root of multiplicity k possible for such a power series. We show that 1 ‐ log ( e k 1 / 2)k + 1 ⩽ r ( k ) ⩽ 1 ‐ 1 k + 1We 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 )∼1− c /( k +1), where c =1.230….