Almirón’s Thom–Sebastiani conjecture for the Tjurina subspectrum

Let $f$ and $g$ be isolated hypersurface singularities, and let $f\oplus g=f(x)+g(y)$ be their separated Thom–Sebastiani sum. Denote by $\operatorname{TjSp}(f)$ the Tjurina subspectrum of $f$ , and by $A\boxplus B$ the multiset obtained by adding every element of $A$ to every element of $B$ . Almirón's conjecture asserts that

\operatorname{TjSp}(f\oplus g)=\operatorname{TjSp}(f)\boxplus\operatorname{TjSp}(g) \quad\Longleftrightarrow\quad \tau_{f\oplus g}=\tau_f\tau_g,

where $\tau_h$ denotes the Tjurina number of the singularity $h$ .

References

Primary source

arXiv

Additional references

Hodge--Tjurina Numbers, Tjurina Spectra, and Thom--Sebastiani Formulas — arXiv — Shijie Zheng, Huaiqing Zuo

algebraic-geometry singularity-theory

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Open

The conjecture remains open: earlier papers give partial information, but no independently checked solution has been found.

Almirón’s conjecture asks whether the Thom–Sebastiani property for the Tjurina subspectrum is equivalent to the multiplicative condition $\tau_{f+g}=\tau_f\tau_g$ . The available literature had posed this equivalence as an open question.

Known results

A 2022 study of Sebastiani–Thom singularities gives formulas for $\tau_{f+g}$ and classifications in cases such as $\tau_{f+g}=\mu_{f+g}-1$ and $\tau_{f+g}=\mu_{f+g}-2$ .
Jung, Kim, Saito, and Yoon (2025) prove graded symmetry for missing spectral numbers and obtain a bound involving $\left\lfloor(\mu-\tau)/2\right\rfloor$ ; this does not resolve the conjecture.

Current status (as of October 2026): The general conjecture remains open; no independently corroborated resolution is recorded.

Sources

No solutions have been posted yet.