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Relation between Hutchinson-invariant and continuously Hutchinson-invariant minimal sets

Ascertain the relationship between the minimal Hutchinson-invariant set (in degree n) and the minimal continuously Hutchinson-invariant set (with parameter ≥ n) for a given linear differential operator T with polynomial coefficients, including identifying conditions under which one set contains or equals the other.

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Background

The paper introduces Hutchinson-invariant sets (defined via images of (x−t)n under T) and continuously Hutchinson-invariant sets (allowing real parameters n ≥ n0), and establishes basic inclusions in some cases (e.g., MH_m ⊆ M{HC}_{≥ m} ⊆ M_{≥ 0}).

However, beyond these inclusions, the authors indicate that the precise relationship between the minimal sets arising from these two notions is not established in general.

References

In general, it is unclear what the relation between ${\geq n}$ and ${\geq n}$ is, but for large $n$, we expect the inclusion ${\geq n} \subseteq ${\geq n}, since extending the domain of $n$ from the set of large integers to the set of large real numbers does not seem to make a big difference.

— An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators (2404.14365 - Alexandersson et al., 22 Apr 2024) in Section 7 (Variations of the original set-up), Variation 3