It has been known since Ehrhard and Regnier's seminal work on the Taylor
expansion of 位-terms that this operation commutes with normalization:
the expansion of a 位-term is always normalizable and its normal form is
the expansion of the B\"ohm tree of the term. We generalize this result to the
non-uniform setting of the algebraic 位-calculus, i.e.
位-calculus extended with linear combinations of terms. This requires us
to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's
techniques rely heavily on the uniform, deterministic nature of the ordinary
位-calculus, and thus cannot be adapted; second is the absence of any
satisfactory generic extension of the notion of B\"ohm tree in presence of
quantitative non-determinism, which is reflected by the fact that the Taylor
expansion of an algebraic 位-term is not always normalizable. Our
solution is to provide a fine grained study of the dynamics of
尾-reduction under Taylor expansion, by introducing a notion of reduction
on resource vectors, i.e. infinite linear combinations of resource
位-terms. The latter form the multilinear fragment of the differential
位-calculus, and resource vectors are the target of the Taylor expansion
of 位-terms. We show the reduction of resource vectors contains the
image of any 尾-reduction step, from which we deduce that Taylor expansion
and normalization commute on the nose. We moreover identify a class of
algebraic 位-terms, encompassing both normalizable algebraic
位-terms and arbitrary ordinary 位-terms: the expansion of these
is always normalizable, which guides the definition of a generalization of
B\"ohm trees to this setting