## Section: New Results

### Complexity of the uncoupling of linear functional systems

Uncoupling algorithms transform a linear differential system of first
order into one or several scalar differential equations. We examined
in [9] two approaches to uncoupling: the
cyclic-vector method (*CVM*) and the
Danilevski-Barkatou-Zürcher algorithm (*DBZ*). We gave tight
size bounds on the scalar equations produced by *CVM*, and
designed a fast variant of *CVM* whose complexity is
quasi-optimal with respect to the output size. We exhibited a strong
structural link between *CVM* and *DBZ* enabling to show
that, in the generic case, *DBZ* has polynomial complexity and
that it produces a single equation, strongly related to the output of
*CVM*. We proved that algorithm *CVM* is faster than
*DBZ* by almost two orders of magnitude, and provided
experimental results that validate the theoretical complexity
analyses.