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Rationality of the Hilbert series of finitely generated Koszul algebras (Polishchuk–Positselski Conjecture)

Prove that for every finitely generated Koszul algebra A, the Hilbert series H_A(t) is a rational function of t.

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Background

The paper recalls a longstanding conjecture by Polishchuk and Positselski concerning Hilbert series of Koszul algebras. This conjecture serves as a foundational motivation for investigating analogous properties for operads. The authors note this conjecture remains unresolved and use it to motivate extensions to symmetric operads and to their own stronger conjecture in the binary-generated operad case.

References

In Conjecture 7.1, Polishchuk and Positselski proposed the following remarkable conjecture: Let $A$ be a finitely generated Koszul algebra, then the Hilbert series of $A$ is a rational function. This conjecture is still open and can be extended to symmetric operads following the insight of Khoroshkin and Piontkovski :

— On Hilbert series of Koszul operads and a classification result for set-operads (2509.14419 - Laubie, 17 Sep 2025) in Introduction (after citing QA, Conjecture 7.1)