Description of the multiplicative Grothendieck group for arbitrary exponents

Describe the multiplicative Grothendieck group $\operatorname{Gr}(\mathcal S)$ for symmetric finitely supported nonnegative combinations of arbitrary real exponents, and determine whether positivity of an element and of all its initial forms on the positive orthant is sufficient for membership in that group.

Background

For commensurable exponents, the paper characterizes membership in the multiplicative Grothendieck group using positivity under all real characters and proves the result through a one-variable form of Pólya’s theorem. The general case involves finitely generated exponent groups of higher rank, where the ambient group ring becomes a multivariate Laurent polynomial ring.

Positivity of a symmetric Laurent polynomial under all characters is necessary but not sufficient. Membership also forces every initial form to be positive on the positive orthant. The paper identifies whether these positivity conditions suffice as a multivariate Pólya-type problem, noting that available toric extensions do not settle it.