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Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280) (1998)
Palo Alto, California
Nov. 8, 1998 to Nov. 11, 1998
ISSN: 0272-5428
ISBN: 0-8186-9172-7
pp: 212
Arnold Schönhage , Universit?t Bonn
ABSTRACT
Let $C_n = C_n(K)$ denote the minimum number of essential multiplications/divisions required for shifting a general $n$-th degree polynomial $A(t)= \sum a_i t^i$ to some new origin $x$, which means to compute the coefficients $b_k$ of the Taylor expansion $A(x+t)= B(t)= \sum b_k t^k$ as elements of $K(x,a_0, \ldots, a_n)$ with indeterminates $a_i$ and $x$ over some ground field $K$. For $K$ of characteristic zero, a new refined version of the substitution method combined with a dimension argument enables us to prove $C_n \ge n + \lceil n/2 \rceil - 1$ opposed to an upper bound of $C_n \le 2n + \lceil n/2 \rceil - 4$ valid for all $n \ge 3$.
INDEX TERMS
CITATION

A. Schönhage, "Multiplicative Complexity of Taylor Shifts and a New Twist of the Substitution Method," Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)(FOCS), Palo Alto, California, 1998, pp. 212.
doi:10.1109/SFCS.1998.743445
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