[Paper Review] A polyhedral approach to computing border bases

Gábor Braun, Sebastian Pokutta|arXiv (Cornell University)|Nov 4, 2009

Polynomial and algebraic computation28 references3 citations

TL;DR

This paper introduces a polyhedral framework for computing border bases of zero-dimensional ideals by characterizing degree-compatible order ideals that support border bases as integral points of the order ideal polytope. The method enables computation of border bases for arbitrary such order ideals via preference-based optimization, overcoming reliance on term orderings and reducing the problem to solving a linear program over the polytope, with the key contribution being a one-to-one correspondence between integral points of the polytope and valid border basis-supporting order ideals.

ABSTRACT

Border bases can be considered to be the natural extension of Gröbner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced Gröbner bases. We adapt the classical border basis algorithm to allow for calculating border bases for arbitrary degree-compatible order ideals, which is \emph{independent} from term orderings. Moreover, the algorithm also supports calculating degree-compatible order ideals with \emph{preference} on contained elements, even though finding a preferred order ideal is NP-hard. Effectively we retain degree-compatibility only to successively extend our computation degree-by-degree. The adaptation is based on our polyhedral characterization: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. This establishes a crucial connection between the ideal and the combinatorial structure of the associated factor spaces.

Motivation & Objective

• To address the open problem of characterizing degree-compatible order ideals that support a border basis for a given zero-dimensional ideal.
• To eliminate dependency on term orderings and reduced Gröbner bases in classical border basis algorithms.
• To enable computation of border bases for arbitrary degree-compatible order ideals, including those with user-defined preferences on monomials.
• To establish a connection between the algebraic structure of ideals and the combinatorial geometry of their factor spaces via polyhedral combinatorics.

Proposed method

• Model the set of degree-compatible order ideals supporting a border basis as the integral points of the order ideal polytope.
• Use linear programming over the order ideal polytope to compute a preferred order ideal based on user-defined monomial weights.
• Adapt the classical border basis algorithm to work with any integral point of the polytope, not just those induced by term orderings.
• Leverage the $L$-stable span procedure as the core computational step, which is polynomial-time and dominates the complexity.
• Formulate the search for optimal order ideals as a mixed-integer program solvable via standard solvers like CPLEX or SCIP.
• Validate the method on multiple polynomial systems, including counting all feasible order ideals and optimizing over preference functions.

Experimental results

Research questions

• RQ1 Which degree-compatible order ideals support a border basis for a given zero-dimensional ideal?
• RQ2 Can the classical border basis algorithm be generalized to work without relying on term orderings or reduced Gröbner bases?
• RQ3 Is it possible to compute a border basis for a preferred order ideal, where certain monomials are prioritized for inclusion?
• RQ4 How can the set of all valid order ideals supporting a border basis be characterized combinatorially and computationally?
• RQ5 What is the computational complexity of finding an optimal order ideal under a given preference function?

Key findings

• There is a one-to-one correspondence between integral points of the order ideal polytope and degree-compatible order ideals that support a border basis for a given zero-dimensional ideal.
• The computation of the $L$-stable span, which dominates the runtime, is significantly more expensive than optimizing over the order ideal polytope, which takes less than 0.1 seconds per run in all tested cases.
• Counting all feasible degree-compatible order ideals was computationally feasible only for smaller instances, with up to 108,900 such ideals found in under 55 seconds.
• For the largest tested system, over 1.3 million order ideals were counted before the 300-second time limit was reached, indicating exponential growth.
• The method successfully computes border bases for arbitrary degree-compatible order ideals, including those with complex preference functions, by reducing the problem to polyhedral optimization.
• The approach enables efficient computation of preferred order ideals even though the underlying optimization problem is NP-hard, due to the dominance of the $L$-stable span computation in overall runtime.