Summation formula for rational powers over tensor products of normal domains

Establish that for any algebraically closed field k, any finitely generated normal k-domains R and S, and T := R ⊗k S, the following Mustață–Takagi-style summation formula holds for rational powers of ideals: for every rational w ≥ 0 and nonzero ideals I ⊂ R and J ⊂ S, one has overline((IT + JT)^w) = ∑_{α ∈ Q, 0 ≤ α ≤ w} overline(I^α) · overline(J^{w−α}) · T.

Background

Rational powers of ideals and Rees valuations are closely connected invariants in commutative algebra and algebraic geometry, but explicit computations are difficult beyond monomial and certain invariant cases. Mustață–Takagi’s summation formula is known for multiplier ideals, and analogous identities have been established for symbolic powers and for rational powers in monomial settings. This conjecture seeks a general summation formula for rational powers across tensor products of normal domains over an algebraically closed field.

The paper proves the conjecture for broad classes of ideals admitting a Rees package (including several invariant and semigroup cases) and establishes an asymptotic version for arbitrary ideals in normal finitely generated k-domains. The full generality, however, remains conjectural as stated here.