Efficient descriptions of univariate polynomial subalgebras and their initial algebras

Abstract For unital and commutative algebras over an algebraically closed field, $$\mathbb {K}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> , any inclusion of finite codimension can be characterised as a chain of subalgebra inclusions of codimension 1. Each such inclusion $$A \subset B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> can also be interpreted as a linear condition that holds on A but not on the whole of B . The set of conditions above A in the chain are called the subalgebra conditions of A . We investigate the behaviour of such chains (or sets of conditions) when the said algebras are ideal subalgebras. That is, when they are sums of the base field and an ideal. We then move to the setting of subalgebras of the univariate polynomial ring, where we can give a more concrete description of subalgebra conditions. We consider a wider class of so-called single-clustered polynomial subalgebras. We show that any subalgebra in $$\mathbb {K}[x]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> of finite codimension is a finite intersection of single-clustered subalgebras. We then present an algorithm to compute subalgebra conditions of single-clustered subalgebras from generators using linear methods. This method is a generalisation of a method used for almost monomial subalgebras, that is, subalgebras with only one element in the spectrum.