Tightness of the quadratic complexity bound

Determine whether the quadratic upper bound in the degree of the input polynomial for the arithmetic complexity of the exact two-dimensional real log canonical threshold algorithm is tight.

Background

The paper presents a deterministic algorithm that exactly computes the local real log canonical threshold of any bivariate polynomial with rational coefficients. Its arithmetic complexity is bounded quadratically in the degree of the input polynomial, through an upper bound on the degree of the finite part of the normalizing power series and consequently on the number of normalization steps.

The discussion explicitly identifies the tightness of this quadratic bound as unresolved. Establishing tightness would clarify whether the algorithm's worst-case complexity reflects an inherent computational difficulty or could be improved asymptotically.

References

We developed an effective algorithm that computes exactly the real log canonical threshold of any input polynomial $f(x,y) \in Q[x,y]$ in a number of steps quadratic in $\deg f$, although the tightness of this bound remains open.

— Exact Algebraic Computation of Learning Coefficients for Two-Dimensional Singular Models  (2608.20183 - Sergeant-Perthuis et al., 20 Aug 2026) in Section 6, Discussion