Multiplicative complexity of Taylor shifts and a new twist of the substitution method

Abstract

Let C/sub n/=C/sub n/(K) denote the minimum number of essential multiplications/divisions required for shifting a general n-th degree polynomial A(t)=/spl Sigma/a/sub i/t/sup i/ to some new origin x, which means to compute the coefficients b/sub k/ of the Taylor expansion A(x+t)=B(t)=/spl Sigma/b/sub k/t/sup k/ as elements of K(x,a/sub 0/,...,a/sub n/) with indeterminates a/sub 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/sub n//spl ges/n+[n/2]-1 opposed to an upper bound of C/sub n//spl les/2n+[n/2]-4 valid for all n/spl ges/3.