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Geometric interpretation of valuated term (pre)orders

Netanel Friedenberg, Kalina Mincheva

TL;DR

This work provides a geometric interpretation for valuated monomial preorders and Chan-Maclagan preorders by linking them to points in a tropical adic space and to prime filters on lattices of polyhedral sets. Using the toric setup with a toric monoid $M$ and value group $\,\Gamma$, the authors establish explicit bijections between Cont$_S S[M]$, Gamma-local equivalence classes of flags of polyhedra, and prime filters on Gamma-rational polyhedral sets containing a polytope. They show that prime congruences correspond to simplicial flags of cones and, modulo local equivalence, these classify CM preorders and valuated preorders, generalizing tropical monomial preorders. The results illuminate a tropical adic analogue of classical bijections in non-archimedean geometry, offering a robust, polyhedral-geometric framework with potential analogues to adic and tropical geometry in broader contexts.

Abstract

Valuated term orders are studied for the purposes of Gröbner theory over fields with valuation. The points of a usual tropical variety correspond to certain valuated terms preorders. Generalizing both of these, the set of all ``well-behaved'' valuated term preorders is canonically in bijection with the points of a space introduced in our previous work on tropical adic geometry. In this paper we interpret these points geometrically by explicitly characterizing them in terms of classical polyhedral geometry. This characterization gives a bijection with equivalence classes of flags of polyhedra as well as a bijection with a class of prime filters on a lattice of polyhedral sets. The first of these also classifies valuated term orders. The second bijection is of the same flavor as the bijections from [van der Put and Schneider, 1995] in non-archimedean analytic geometry and indicates that the results of that paper may have analogues in tropical adic geometry.

Geometric interpretation of valuated term (pre)orders

TL;DR

This work provides a geometric interpretation for valuated monomial preorders and Chan-Maclagan preorders by linking them to points in a tropical adic space and to prime filters on lattices of polyhedral sets. Using the toric setup with a toric monoid and value group , the authors establish explicit bijections between Cont, Gamma-local equivalence classes of flags of polyhedra, and prime filters on Gamma-rational polyhedral sets containing a polytope. They show that prime congruences correspond to simplicial flags of cones and, modulo local equivalence, these classify CM preorders and valuated preorders, generalizing tropical monomial preorders. The results illuminate a tropical adic analogue of classical bijections in non-archimedean geometry, offering a robust, polyhedral-geometric framework with potential analogues to adic and tropical geometry in broader contexts.

Abstract

Valuated term orders are studied for the purposes of Gröbner theory over fields with valuation. The points of a usual tropical variety correspond to certain valuated terms preorders. Generalizing both of these, the set of all ``well-behaved'' valuated term preorders is canonically in bijection with the points of a space introduced in our previous work on tropical adic geometry. In this paper we interpret these points geometrically by explicitly characterizing them in terms of classical polyhedral geometry. This characterization gives a bijection with equivalence classes of flags of polyhedra as well as a bijection with a class of prime filters on a lattice of polyhedral sets. The first of these also classifies valuated term orders. The second bijection is of the same flavor as the bijections from [van der Put and Schneider, 1995] in non-archimedean analytic geometry and indicates that the results of that paper may have analogues in tropical adic geometry.
Paper Structure (8 sections, 36 theorems, 38 equations, 4 figures)

This paper contains 8 sections, 36 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Flags and prime congruences.
  5. Equivalence of flags.
  6. The filter of a prime congruence.
  7. Flags and filters.
  8. From filters back to congruences.

Key Result

Theorem A

There is an explicit bijection from the set of valuated monomial preorders on $k[x_1^{\pm1},\ldots,x_n^{\pm1}]$ to the set of $\Gamma$-local equivalence classes of flags of polyhedral cones in $\mathbb{R}_{\geq0}\times\mathbb{R}^n$.

Figures (4)

  • Figure 1: The flags $\mathcal{P}_\bullet$ and $\mathcal{P}_\bullet'$ of Example \ref{['ex:Whale/Dolphin']}.
  • Figure 2: This picture illustrates the filters corresponding to $\mathcal{P}$ and $\mathcal{P}'$ by drawing the "infinitesimal neighborhoods" that a rational polyhedron must contain to be a neighborhood of the flags. The turquoise represents the condition to be a neighborhood of the truncation $\mathcal{P}_\bullet^{(0)}=(\mathcal{P}_\bullet')^{(0)}$; the purple represents the added conditions to be neighborhoods of $\mathcal{P}_\bullet$ and $\mathcal{P}_\bullet'$.
  • Figure 3: We see that the filters are equal, so the flags $\mathcal{P}_\bullet$ and $\mathcal{P}_\bullet'$ are locally equivalent.
  • Figure 4: The flags of polyhedra and filters from Example \ref{['ex:JellyfishAndFriends']}. In these pictures, turquoise represents "one-dimensional infinitesimal neighborhoods" and purple represents "two-dimensional infinitesimal neighborhoods".

Theorems & Definitions (88)

  • Theorem A: Corollary \ref{['coro:explicit BijectionsAllPrimeCongsAndThreeOthers']}
  • Theorem B: Corollary \ref{['coro:explicitBijectionsContAndThreeOthers']}
  • Theorem C: Proposition \ref{['prop:whenDoesAPrimeTotallyOrder']}
  • Theorem D: Theorem \ref{['thm:BijectionFiltersAndCont']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Congruences
  • Definition 2.5
  • Lemma 2.7: FM22 Lemma 2.12, Downward gaussian elimination
  • ...and 78 more