Describing subalgebras of $\mathbb{K}[x]$ using derivatives

We introduce the concept of subalgebra spectrum, $Sp(A)$, for a subalgebra $A$ of finite codimension in $\mathbb{K}[x]$. The spectrum is a subset of the underlying field. We also introduce a tool, the characteristic polynomial of $A$, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of $A$, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that $A$ can be described by a set of conditions that each is either of the type $f(α)=f(β)$ for $α,β$ in $Sp(A)$ or of the type stating that some sum of derivatives of different orders evaluated in elements of $Sp(A)$ equals zero. We use this type of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.