Suppose, $a_1, a_2, \ldots, a_n$ are distinct points in a finite field $\mathcal{F}$ whose size is bigger than and independent of $n$. Also, $P_0(x), P_1(x),\ldots, P_n(x)$ are randomly chosen $t$ $(<n)$ degree polynomials over $\mathcal{F}$. Does it necessarily imply that the cardinality of the set $\mathcal{T}$ $=$ $\{P_0(x) + a_iP_1(x) + {a_i}^2P_2(x) + \cdots + {a_i}^nP_n(x) \mid i = 1,2,\dots, n \}$ is $n$?
Intuitively, I think the cardinality is not, in general, $n$, given the fact that the polynomials are randomly chosen but neither I could prove the statement nor find a counterexample.
Please, consider the finite field to be of prime characteristic. This eventually leads to the fact that it is bit difficult to provide a specific example but it MAY be possible to provide a 'hunch' or intuition about the counterexample. That much will do.
Also, I would like to request to consider $n$ to be at least more than 100.
(I found some questions in the forum addressing the nature of roots or linear combinations of polynomials over a finite field but none of them answers this.)