Zero Spectrum Subalgebras of K[x] Described by Higher Derivatives

Abstract: Unital subalgebras of finite codimension in the polynomial ring $\mathbb{K}[x]$ are described by a finite number of so called subalgebra conditions over a finite set in $\mathbb{K}$ named the subalgebra spectrum. Restricting attention to subalgebras whose spectrum is the singleton $\{0\}$ reveals a rather well behaved class of subalgebras, called almost monomial from the fact that these contain an ideal consisting of all monomials above a certain degree. Analysing them is made smoother with the introduction of the lower degree of a polynomial, giving rise to a lower numerical semigroup and, in turn, a linear basis consisting of a finite number of basis vectors from a quotient along with all monomials of sufficiently large degrees. The main result of this thesis is that the subalgebra conditions of almost monomial algebras are found from the annihilator of this quotient. Hence the subalgebra conditions are found from solving a matrix kernel problem, given a linear basis of this particular kind. By applying this result to one of the simplest kind of almost monomial subalgebras, a proof for the existence of a previously conjectured sequence of derivations is revealed.