When thinking about monoid algebras (semigroup rings), I came across the following:
Let $S$ be a numerical semigroup, i.e., an additively closed subset of $\mathbb{N}_0$ containing $0$ such that $\mathbb{N}_0 \setminus S$ is finite. Let $K$ be any field.
Question: Do there exist polynomials $a, h \in K[x]$ with $a(0) = h(0) = 1$ and $a$ has only exponents in $S$ such that their product $ah$ has only exponents in $(\mathbb{N}_0 \setminus S) \cup \{0\}$.
Say, $a=1+k_{s_1}x^{s_1}+\ldots +k_{s_n}x^{s_n}$. Then it is immediate, that $h$ has to contain the monomial $-k_{s_1}x^{s_1}$. It is also clear, that the largest exponent of $h$ cannot be in $S$. I have the impression that the above question can be answered negatively, but I just do not know how to prove it.
Has anyone an idea how to prove this?
Thank you in advance for you help!