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The function $f(k)$ is a (numerical) polynomial in $\mathbb{Q}[k]$, and the set $ S_f = \{ f_d : d \in \mathbb{N} \} $ is a set associated with $ f $, where $f_d(k)=f(k+d)$.

I am interested in finding invariants related to $S_f$, that is, I want to find a map from $\{ S_f \}$ to some set, such as $\deg(f)$ and the leading coefficient of $ f $. Note that it is not necessarily a map from $\{ S_f \}$ to $\mathbb{Q}$; it would be great if each given $S_f$ could correspond to an algebraic or geometric structure.

By comparing coefficients, one can directly obtain some invariants modulo the coefficients of higher-order terms, but I am looking for more "interesting" conclusions.

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  • $\begingroup$ Well, every such set contains a unique polynomial starting with $a_n k^n + a_{n-1}k^{n-1}+\ldots$ with $0 \le a_{n-1} < da_n$, so you can map $S_f$ to this polynomial. $\endgroup$
    – Aleksei Kulikov
    Commented 6 hours ago
  • $\begingroup$ By $\{S_f\}$ do you mean $\{S_f:f\in Q[x]\}$? You really want a map from this set, or from the set of polynomials? What is $k$ in the definition of $S_f$? $\endgroup$
    – YCor
    Commented 5 hours ago
  • $\begingroup$ @YCor, I want a map from this set. In other words, I want a map $\phi$ from $\mathbb Q[k]$ such that $\phi(f) =\phi(f_d)$ for each $d\in \mathbb N$, where $f_d:k\mapsto f(k+d)$. $\endgroup$
    – zhjzwlys
    Commented 4 hours ago

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