Erik Kennerland presents his Bachelor thesis in mathematics: "Zero spectrum subalgebras of K[x] decribed by higher derivatives"

Abstract:

Unital subalgebras of finite codimension in the polynomial ring K[x] are described by a
finite number of so called subalgebra conditions over a finite set in K named the subal-
gebra 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 polyno-
mial, 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.