There's a well-known result of Cherlin, Harrington, and Lachlan that gives a very precise structural understanding of $\aleph_0$-categorical, $\aleph_0$-stable theories. One thing in particular is that strongly minimal sets in such theories must be (finite covers of) either disintegrated sets or affine or projective spaces over finite fields.
I'm curious about the number of parameters that are necessary to define strongly minimal sets in such theories and, in particular, whether the geometry of the strongly minimal set matters for this. It's very easy to construct an example where a single parameter is needed to specify a strongly minimal set, the theory of an equivalence relation with infinitely many infinite equivalence classes, for instance. It's easy to modify this example to give the strongly minimal sets any of the allowed geometries.
But beyond this, I can't see how to construct an example with strongly minimal sets requiring more than one parameter. Specifically, in everything I can think of, there's always a $\varnothing$-definable equivalence relation whose equivalence classes are strongly minimal sets. So this motivates my questions.
EDIT: For the sake of making this question non-trivial, when I say that a strongly minimal set $\varphi(x)$ essentially requires $n$ parameters to define, I mean that there is a strongly minimal set $\psi(x)$ definable with $n$ parameters such that the symmetric difference of $\varphi(x)$ and $\psi(x)$ is finite, and there is no such set definable with fewer than $n$ parameters. This is the same as talking about the number of parameters needed to define the corresponding strongly minimal type.
Question 1: For which $n$ are there $\aleph_0$-categorical, $\aleph_0$-stable theories with strongly minimal sets that essentially require at least $n$ parameters to define?
Question 2: For which $n$ are there such theories where every strongly minimal set essentially requires at least $n$ parameters to define?
Question 3: Does the answer change if we require that the strongly minimal sets have a certain geometry?
One way that there might be a bound on such $n$ is if the situation with the equivalence relation actually always happens. This raises another natural question.
Question 4: Does an $\aleph_0$-categorical, $\aleph_0$-stable theory always have a $\varnothing$-definable equivalence relation whose equivalence classes are all either strongly minimal or finite?
A less trivial example of this is the theory of $(\mathbb{Z}/p^k\mathbb{Z})^\omega$, which has the equivalence relation defined by $p^{k-1} x = p^{k-1} y$.
There's some precedent for bounds like this. Specifically, I vaguely recall a result of Hrushovski that implies that if there's a finite-to-finite correspondence between two strongly minimal sets, then this correspondence can require at most 3 parameters to define (beyond the parameters necessary to define the strongly minimal sets in the first place). Furthermore, the number of parameters required is closely related to the geometry. Affine spaces can require 1, projective spaces can require 2, and projective lines (over an algebraically closed field) can require 3.
