Almost monomial subalgebras of $$\mathbb {K}[x]$$ and their LAGBI bases

We investigate subalgebras of finite codimension in $$\mathbb {K}[x]$$ . In earlier work we have introduced a way of describing such subalgebras in terms of their so called (subalgebra) spectrum and a set of conditions for subalgebra membership that can be expressed by evaluating polynomials and their derivatives in points of the spectrum only. In this paper we focus on subalgebras with a single element in their spectrum. This includes, among others, all monomial subalgebras. Moreover, any subalgebra given by only conditions involving derivatives can be obtained as a finite intersection of algebras with single spectrum. Our main result is an efficient algorithm for finding the set of defining conditions given a set of generators for a single spectrum subalgebra. As an important step on the way to an algorithm we introduce a new canonical basis (with many similarities to SAGBI basis), that we name LAGBI basis, for our single spectrum algebras. We then find an efficient algorithm for computing a LAGBI basis and finally incorporate it into our main algorithm for finding defining conditions. In the process we also find the derivations of a single spectrum subalgebra.