Characterization of good coordinate systems in dimensions greater than two

Characterize coordinate systems that are well adapted to Newton diagrams for computing real log canonical thresholds of analytic functions or polynomials in dimensions greater than two.

Background

The algorithm developed in the paper is restricted to biparametric models, corresponding to functions of two variables. Its effectiveness relies on the two-dimensional theory of Newton polygons and on normalizing changes of variables that can be described using univariate Puiseux expansions.

The paper states that, beyond two dimensions, a fundamental unresolved issue is the characterization of coordinate systems suitable for Newton-diagram methods. Resolving this question would be a prerequisite for extending the exact algorithmic framework to higher-dimensional models; the paper mentions multivariate fractional power series as one possible direction.

References

As observed in \citep[Section~III]{phong1999growth}, in dimensions $>2$ we still do not know how to characterize good coordinate systems with respect to Newton diagrams.

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