Finite generation and effective generators over a polynomial ring

Determine whether the quotient of the admissible forcing cone by its lineality module is finitely generated as a semimodule over the nonnegative restricted polynomial ring when the coefficient matrix is univariate polynomial, and characterize additional algebraic conditions under which finite generation holds and finite generators can be computed effectively from the matrix and differentiability order.

Background

The paper establishes finite generation for the lineality module over both restricted polynomial rings and Nash-function rings, but shows that the quotient semimodule can fail to be finitely generated over a restricted polynomial ring when the matrix contains a nonrational Nash entry.

The polynomial-matrix case remains unresolved. The question asks both whether finite generation always holds for polynomial matrices and which additional algebraic hypotheses would ensure it, together with an effective method for computing generators. The authors note that this problem may require algebraic methods beyond the selection and slack-variable techniques developed in the paper.