Algebraic-function formulation of Hilbert’s thirteenth problem

Determine whether the algebraic-function version of Hilbert’s thirteenth problem can be solved constructively using exp-log expressions represented by EML operators.

Background

The paper distinguishes its notion of elementary functions—finite expressions formed from standard scientific-calculator operations—from the narrower differential-algebraic notion of elementary functions. Under the latter interpretation, expressing algebraic functions such as roots of generic high-degree polynomials would amount to solving Hilbert’s thirteenth problem in an exp-log framework.

The authors explicitly state that the algebraic-function version remains unresolved, while clarifying that this is not claimed as a consequence of the original EML construction.

References

So if one interpreted title of in a narrow differential-algebraic strictly mathematical sense, then EML expression would provide e.g. constructive exp-log solution $f(a,b,c)$ to Hilbert's 13th problem, which involves roots of the 7-th order polynomial $ x7 + a\, x3 + b\, x2 + c \, x +1 =0$ with 3 parameters $a,b,c$. This is, of course, not what claims, and the algebraic-function version (cf. Subsect.~\ref{prior}) of the Hilbert's 13th remains open.

— Diversity of EML-type operators  (2609.11210 - Odrzywołek, 10 Sep 2026) in Section 2.1, “All elementary functions?”

However, it is not clear how to extend hash to exp-log functions in any other way than by extending the hash operator with $\exp(x)$ and $\ln(x)$ themselves.

— Diversity of EML-type operators  (2609.11210 - Odrzywołek, 10 Sep 2026) in Section 2.7, “Prior knowledge”