Choose principled monomial orders for completion and degree bounds

Determine criteria or algorithms to select a semigroup (monomial) order on monomials that optimizes properties of Norman’s completion process and its refinement (Algorithm 3) for constructing complete reduction systems and computing degree bounds for linear operators L defined by Eq. (16); characterize how order selection affects termination of the completion process and the admissible weight vectors for rigorous degree bounds derived via Theorem 40.

Background

The paper shows that termination of both Norman’s original completion process and the refined process depends on the chosen monomial order, and that the order also constrains feasible weight vectors when deriving degree bounds from reduction systems.

The authors explicitly highlight the selection of monomial orders as an open problem because it materially impacts algorithmic termination and the tightness and applicability of the resulting bounds.