Effective computation of Nash lineality generators and quotient relations

Determine whether finite generators for the lineality module of admissible forcing vectors over the Nash-function ring on a regular interval can be computed effectively from the univariate semialgebraic matrix and the differentiability order, and whether the relations among the coordinate classes can likewise be computed effectively so as to produce a minimal semimodule generating set for the quotient by lineality.

Background

For a matrix of univariate semialgebraic functions and a regular interval, the paper proves that the lineality module of admissible Nash forcings is finitely generated over the Nash-function ring. The quotient of the admissible forcing cone by this lineality module is generated as a semimodule by the coordinate classes.

The unresolved issue is algorithmic rather than existential: the paper does not provide an effective procedure for computing generators of the lineality module or the relations among the coordinate classes. Such a procedure would yield a minimal semimodule generating set for the quotient and would extend known algorithmic results for systems of linear equations to systems of inequalities.

References

Can a finite set of generators for the lineality module ${A,I}m((I)) $ be computed effectively from $A$ and $m$? Can one also effectively compute the relations among the coordinate classes \begin{equation*} [e_1],\ldots,[e_N] \in (I)N\Big/{A,I}m((I)), \end{equation*} so that one can obtain a minimal set of generators for ${A,I}m((I)) \Big/ _{A,I}m((I))$ as a semimodule over $+(I)$?

— Semialgebraic ${C^m}$ solutions to a univariate system of linear inequalities with Nash forcing  (2610.22772 - Jiang et al., 16 Oct 2026) in Question (A), Section "A subsequent algebraic question"