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Order-one differential-algebraicity conjecture for binary-generated Koszul symmetric operads

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Let P\mathcal{P} be a Koszul symmetric operad generated by one element in arity 22, and let fPf_{\mathcal{P}} denote its Hilbert series. Being differential algebraic of order 11 means that a differential-algebraic relation involves only fPf_{\mathcal{P}} and its first derivative fP′f_{\mathcal{P}}'.

Order-one differential-algebraicity conjecture. The Hilbert series of P\mathcal{P} is differential algebraic of order 11 over Z[t]\mathbb{Z}[t].

Equivalently, fPf_{\mathcal{P}} and fP′f_{\mathcal{P}}' are algebraically dependent over Z[t]\mathbb{Z}[t]. The conjecture is motivated by the known examples of Koszul symmetric operads generated by one binary operation, whose Hilbert series satisfy differential-algebraic identities of order 11; its general status is not specified in the supplied text.

References

Primary source

Paul Laubie, “On Hilbert series of Koszul operads and a classification result for set-operads”, arXiv:2509.14419 (2025).

hilbert-serieskoszul-operadssymmetric-operads
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Progress summary

Refreshed
Claimed progress

A 2025 paper settles the conjecture for eleven restricted set-operads, but the full question for symmetric operads remains open.

The conjecture asserts that every Koszul symmetric operad generated by one binary operation has a Hilbert series whose value and first derivative satisfy an algebraic relation over Z[t]\mathbb{Z}[t]. The supplied sources identify this as an open conjecture, not a theorem for all such operads.

Known results

  • All 1111 Koszul symmetric set-operads generated by one binary operation satisfy the order-11 statement; the classification includes 44 new cases.

September 2025 classification result

The relevant paper claims a complete classification of the narrower set-operad case and derives differential algebraicity of order 11 for every member. It does not classify all Koszul symmetric operads or resolve the stated conjecture. No counterexample, complete proof, or later verification was found.

Current status (as of September 2026): The conjecture remains open for general Koszul symmetric operads, while the order-11 claim is reported for all 1111 classified set-operads.

Sources

Solutions 0

No solutions have been posted yet.