Beukers relation basis with a polynomial left inverse

For any finite family of complex formal power series $f_1,\ldots,f_m$, there exist polynomial matrices $C$ and $U$ such that the rows of $C$ generate exactly all polynomial relations among the $f_i$, and

Consequently the relation rows are independent over $\mathbb C[X]$, and their specializations remain independent at every complex number.

This is a complete proof of the relation-basis ingredient in Beukers lifting. The proof uses that the image of the relation map is a finite torsion-free module over the PID $\mathbb C[X]$, hence free and projective. Splitting the map gives a retraction onto its kernel and hence the displayed left inverse. No E-function arithmetic or transcendence theorem is assumed.