Ax–Schanuel Theory for Non-Isoconstant D-Groups

Definitions and mathematical statement

Let $(K,\partial)$ be a characteristic-zero differential function field, with constant field $C=\ker\partial$, in the intended complex setting $C={\mathbb{C}}$. The supplied record does not fix the base variety, number of derivations, or differential overfield. For a smooth commutative algebraic group $G/K$, an algebraic $D$-group structure is a compatible algebraic group section $s:G\to\tau_\partial G$ of the differential prolongation. If $(L,\partial)\supseteq(K,\partial)$ is a differential extension, the difference between the derivative of $y\in G(L)$ and $s(y)$, translated to the identity, gives the logarithmic derivative \[ \partial\log_G:G(L)\longrightarrow\operatorname{Lie}(G)(L). \] The induced $D$-structure on the Lie algebra gives a differential operator $\partial_{\operatorname{Lie}(G)}$. Its horizontal vectors and the solutions of $\partial\log_G(y)=0$ are the respective constant-solution objects.

In this language, the requested extension of Ax's generalized Schanuel theorem concerns transcendence bounds and differential-Galois descriptions for logarithmic and exponential equations, schematically \[ \partial\log_G(y)=a, \qquad \partial\log_G(y)=\partial_{\operatorname{Lie}(G)}(x). \] The problem asks to extend the constant commutative algebraic-group theory over ${\mathbb{C}}$ to non-isoconstant algebraic $D$-groups over a function field, including both formulations and their differential-Galois interpretation. For a constant group $G_0/C$, base change to $K$ with its constant $D$-structure supplies the reference constant case. Under the differential-group meaning, isoconstant means becoming isomorphic to such a constant $D$-group after an allowed differential field extension; if the source instead uses isoconstancy of the underlying algebraic group, that distinction must be resolved.

This is a research-program request, not a fully specified quantified inequality in the supplied record. In particular, it supplies no numerical transcendence-degree bound, base field relative to which it is measured, allowed class of nonconstant groups, or independence condition modulo proper algebraic or $D$-subgroups. No claim that every solution of either displayed schematic equation is generic is intended.

Short English statement

The requested theory would extend the constant-group Ax transcendence framework to varying algebraic differential groups over function fields, with logarithmic, exponential, and differential-Galois versions. The record leaves the precise conjectural bounds and nondegeneracy hypotheses unstated.

Sources

• Local UnsolvedMath registry, permanent ID $30000768$, exact source code catalogue entry, proposed year 2007; no individual proposer is supplied. The requested extension is attributed to the framework of Ax's generalized Schanuel theorem.
• Arithmetic and Differential Galois Groups (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/26, pp. 1443--1520. https://doi.org/10.4171/owr/2007/26.
• The supplied corrected_verified note reports editorial and LaTeX cleanup without added source-dependent content. The open label and assessment checked 2026-08-21 say Ax's theorem settles the isoconstant algebraic-group case, while the proposed non-isoconstant extension and differential-Galois interpretation remain open. This is dated supplied triage, not independent verification of its scope or current status.