In previous work we have shown that any subalgebra \(A \subseteq \mathbb {K}[x]\) of finite codimension n can be described by a finite set \(\text {Sp}(A)\) , called the subalgebra spectrum of A, together with n defining conditions. In this paper we show that a similar result applies to subalgebras of infinite codimension. We also develop new tools for the case of finite codimension. One of them is to introduce an ideal I(A) in A of finite codimension in \(\mathbb {K}[x]\) , such that \(V(I(A))=\text {Sp}(A).\) We use A/I(A) to analyse how the structure of A relates to its defining conditions. A key problem is to find algorithms for obtaining a SAGBI basis given defining conditions and vice versa. We present an efficient solution to the first problem. By elimination of a linear system we obtain a linear basis of A and then reduce it into a minimal SAGBI basis. By solving the same linear system we can also obtain the normal form of any polynomial. Further, we suggest an efficient way to obtain \(\text {Sp}(A)\) from generators of A, but note that finding the conditions is NP hard.
