Inc-Compatible Term Orders and Gröbner Theory in Infinitely Many Variables

Time: 9:30 - 11:00, July 08, 2026

Venue: Room 301, A5, Institute of Mathematics-VAST

Abstract: Polynomial rings in infinitely many variables arise naturally in algebraic statistics, combinatorial algebra, and the study of equivariant Gröbner bases. A fundamental challenge in this setting is the construction of monomial orders that are compatible with the action of the monoid of increasing maps, commonly denoted by Inc. Such compatibility is essential for extending Gröbner basis techniques from finite-dimensional polynomial rings to infinite-variable settings.

In this talk, we present a systematic study of Inc-compatible term orders and develop foundational results toward a general theory of these orders. We discuss necessary and sufficient conditions for Inc-compatibility, investigate structural properties of admissible term orders, and provide constructions and examples illustrating the theory. The results contribute to the understanding of equivariant Gröbner methods and offer new tools for studying ideals and algebraic structures possessing infinite symmetry.