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

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)=Σa_itⁱ to some new origin x, which means to compute the coefficients b_k of the Taylor expansion A(x+t)=B(t)=Σb_kt^k as elements of K(x,a₀,...,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⩾n+[n/2]-1 opposed to an upper bound of C_n⩽2n+[n/2]-4 valid for all n⩾3