The monomial-ordering transcendence-degree conjecture

Let $R$ be a ring equipped with a monomial ordering

preceq$, and let the statements denoted by Theorem~tMain, equations~(tMainA) and~(tMainC) concern the transcendence degree

trdeg of $R$. Monomial-ordering transcendence-degree conjecture. The statements denoted by TheoremtMain, equations(tMainA) and~(tMainC), hold with

$$trdeg replaced by$$

trdeg_\preceq$, for an arbitrary monomial ordering

preceq$. This conjecture is motivated by computations for triples of polynomials in

mathbb Z[x]$ under graded reverse lexicographic ordering, but its general validity for arbitrary rings and monomial orderings is left open.