# How do I find non-linear sets that are invariant under a certain linear transformation?

I have an invertible linear transformation $T:F^k\to F^k$, where $F$ is a finite field and $k$ is a natural number. It's easy to find the linear subspaces S that are invariant under T. How do I find non-linear sets S (e.g., sets defined by degree-2 multivariate polynomials $\{ (p_1(t_1,...,t_n),p_2(t_1,...,t_n)...)|t_1,...,t_n \in F \}$ that are invariant under T?

The context in which this question arose in my research: I'm looking for a certain construction in which sets have to satisfy two properties: one is invariance under $T$, the other is a property that is pretty well understood (the vectors in the set can be thought of as the rows of a generating matrix of a linear error correcting code with good parameters).

Any information about invariance of non-linear sets (e.g., forms of $T$ from which its non-linear invariant sets could be easily deduced, pointers to relevant math theorems, etc) might be useful for me.

• Without further conditions on $T$ or some maximality condition on your non-linear set $S$, this seems tricky to answer sensibly. (Case in point: take T to be the identity and S to be anything very like a whale.) Is there a more precise version of this question, specifying domains etc, that you coudl pose? Aug 18 '11 at 1:39
• What I'm looking for is a reference to the mathematical area/technique that deal with such problems. Aug 18 '11 at 1:49
• But what kinds of set are you looking for? It seems like you want algebraic sets of some kind (e.g. solutions to systems of equations) and not just any old set. Aug 18 '11 at 2:28
• It's clear isn't it? Let $V$ be the $+1$-eigenspace (which might be $0$) of $T$. Then any subset of $V$ is invariant. Conversely, the linear span of an invariant set would have to lie in $V$. Aug 18 '11 at 2:42
• Or do you mean sets such that $T(S)\subset S$? Aug 18 '11 at 2:51