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A numerical semigroup $A$ is defined as a subsemigroup of the semigroup $(\mathbb{N},+)$ of the positive integers such that the set $\mathbb{N}\setminus A$ is finite. Equivalently (for a subsemigroup) this means that $\gcd(A)=1$. Moreover, every subsemigroup of $(\mathbb{N},+)$ is finitely generated.

I would like now to ask whether the following kind of assertion is known (I couldn't find any hints on the internet so far):

For a numerical semigroup $A$, generated by the (ordered) set $S=\{a_1<\dots<a_n\}$, define a polynomial $h_{S}=x^{a_1}+\cdots+x^{a_n}\in\mathbb{Z}[x]$.

(On the other hand, given a polynomial $h\in\mathbb{Z}[x]$ with zero constant term and coefficients in $\{0,1\}$ set $S_{h}=\{k|\ x^{k}\ \text{appears in}\ h \}$ and in the case that $\gcd(S_{h})=1$ let $A_{h}$ be the (numerical) semigroup generated by $S_{h}$).

Let $\mathrm{ilt}(f)$ for any non-zero polynomial in $\mathbb{Z}[x]$ denote the initial term (of the lowest degree) of $f$ (e.g. for $f=3x^2 + x^5$ is $\mathrm{ilt}(f)=3x^{2}$).

Now define (for a numerical semigroup $A$ generated by $S$) the following recurrent sequence of polynomials:

$h_{0}=h_{S}$

$h_{n+1}=h_{S}\cdot \mathrm{ilt}(h_{n})+(h_{n}-\mathrm{ilt}(h_{n}))$.

Let $N=\deg(h_{S})$. The assertion now is, that for every $K>0$ there is $n_{0}\in\mathbb{N}$ such that for every $n\geq n_{0}$ we have $S_{h_{n}}= A\cap\{\deg(h_{n})-N+1, \deg(h_{n})-N+2,\dots,\deg(h_{n})\}$ and all the (non-zero) coefficients of $h_{n}$ are bigger than $K$.

One can also show that every element of $A$ appears as the exponent in the initial term of some $h_{n}$.

Because of this I would expect that such a property should be known (and probably there is some terminology for it e.g. "a characteristic polynomial of a numerical semigroup" or similarly).

I will be grateful for any hint.

EDIT: A typo $N=\deg(h_{s})$ fixed to $N=\deg(h_{S})$. (Thanks to Gerhard Paseman). A bit longer answer to Gerhard now follows:

@Gerhard: You're right. My motivation for this question is to use semigroups for solving problems about polynomials (more precisely about semirings). Let $\mathbb{N}[x]$ denote the semiring of all non-zero polynomials with non-negative integer coefficients over the variable $x$. The assertion in my question is now used to show that:

For a semiring $T$ that is a homomorphic image of $\mathbb{N}[x]$ the following are equivalent:

(i) $T+T=T$ and there are $a,b\in T$ such that $a=a+b$.
(ii) There is $c\in T$ such that $1=1+c$.

If $w\in T$ is the corresponding generator of $T$, the proof of this claim splits into two cases when $Tw\neq T$ (which is easy) and when $Tw=T$. Now the condition $Tw=T$ is equivalent to $1=h(w)$ for some polynomial $h\in\mathbb{N}[x]$ with a zero constant term and I use the assertion from my question to construct the desired equation $1=1+c$.

Such a claim motivated also my other question: Terminology for the equation $a=a+b$ in commutative semigroups .

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  • $\begingroup$ I don't know what $h_s$ is, so am unsure about $N$. Also, $h_n$ may not be of the right shape to fit your definition of $S_{h_n}}$, so one of them may have to bend. While encoding information about $A$ in polynomials may be useful, unless you use polynomials to solve problems about $A$ or semigroup results to solve problems about $h$, I don't see the point. Can you suggest more motivation? Also, does a web search for "polynomial encoding of numerical semigroups" help? Gerhard "Ask Me About System Design" Paseman, 2014.03.26 $\endgroup$ Mar 26, 2014 at 20:16

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