A priori structural identifiability analysis of a system of ordinary differential equation using differential algebra

Question

Consider an ordinary differential equation model of a dynamic system:

$\dot{x} = f(x,u,p)$

$y = g(x,p)$

where $x$ is the n-dimensional state vector, $u$ is the r-dimensional input vector, $p$ is the p-dimensional parameter vector, and $y$ is the m-dimensional output vector.

The differential algebra approach for a priori structural identifiability analysis is as follows:

1. Reduce the aforementioned ODE model into input-output map of the form $\psi(u,y,p)=0$ using Ritt's pseudodivision algorithm. The result is called the characteristic set.
2. Decompose the input/output map as a linear combination of the polynomials from a certain differential ring $\mathbb{R}(p)[u,y]$. In other words, $\psi(u,y,p) = \displaystyle\sum c_i(p) \psi_i(y,u)$.
3. If the ideal generated by the characteristic set is prime, then $\psi_i(y,u)$ are linearly independent.
4. Global identifiability can then be analysed by forming an injectivity condition $\psi(u,y,p) = \psi(u,y,p*)$, which becomes $\displaystyle\sum (c_i(p)-c_i(p*)) \psi_i(y,u) = 0$. Global identifiability then becomes injectivity of the map $c(p)$.

However, from the linear independence of $\psi(y,u)$ , I would also infer that $c_i$'s in the above expression are all zero.

Since $c_i$'s obviously can't all be zero, I think I am missing something very fundamental here. Any ideas on what I might be missing would be very helpful.

Please refer to Section 2, Section 3, and Section 4 of https://www.sciencedirect.com/science/article/abs/pii/S0025556409001485?via%3Dihub to understand the above approach in more detail.

Thank you.