A Polyhedral Characterization of Border Bases

Abstract

A. Border bases arise as a canonical generalization of Gröbner bases, using order ideals instead of term orderings. We provide a polyhedral characterization of all order ideals (and hence all border bases) that are supported by a zero-dimensional ideal: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. In particular, we establish a crucial connection between the ideal and its combinatorial structure. Based on this characterization we also provide an adaptation of the border basis algorithm of Kehrein and Kreuzer [35] to allow for computing border bases for arbitrary order ideals, given implicitly via maximizing a preference on monomials (variable selection problem), independent of term orderings. The algorithm requires the same size of resources as the border basis algorithm except for some minor overhead. We also show that the underlying variable selection problem of finding an order ideal that supports a border basis is NP-hard and that any linear description of the associated convex hull of all order ideals requires a superpolynomial number of inequalities. IIn many different disciplines and real-world applications one is faced with solving systems of polynomial equations. Often this is simply due to a physical or dynamical system having a natural representation as a system of polynomial equations, but equally often it is due to the sheer expressive power of polynomial systems of equations that allow for easy reformulation. To give an example of the latter, an inequality ax ≤ b with a ∈ R n and b ∈ R can be expressed via a single polynomial equation: ax + u 2 = b. A slightly more involved example is that of the feasible region of a binary program {x | Ax ≤ b, x ∈ {0, 1} n }, which can be captured via rewriting each individual inequality as before, and adding quadratic polynomials x 2 i − x i = 0 for each coordinate i = 1, . . . , n of x. As a consequence, there is a huge need to computationally model, understand, manipulate, and extract the solution set of systems of polynomial equations.A key insight in (computational) commutative algebra is that one can choose a smart ordering on the monomials and compute a special set of generators of the ideal generated by the system of equations that makes many operations easy, and provides a structural insight into the system. One such special set of generators is a Gröbner basis. By now, Gröbner bases are fundamental and standard tools in commutative algebra to actually perform important operations on ideals such as intersection, membership test, elimination, projection, and many more. Border bases arise as a natural generalization of Gröbner bases that can be computed for zero-dimensional ideals, i.e., the associated factor space is a finite-dimensional vector space (see Section 2 for details). While this might seem to be a severe restriction, for many applications it is sufficient. Roughly speaking, whenever the solution set is finite, we are dealing with a zero-dimensional ideal. For example, systems of p...

Computing all border bases for ideals of points

“…[2] Given an order ideal O ⊂ T and an ideal I ⊂ R, we say that I supports an O-border basis if the residue classes of the terms in O form a basis of R/I as a -vector space. [3] If O ⊂ T is an order ideal, the set…”

“…Thus we are willing to pay the cost of computing repeated results and remove them later. Finally, we draw the attention of the reader to the example X = {(0, 0, 0, 1), (1, 0, 0, 2), (3, 0, 0, 2), (5, 0, 0, 3), (−1, 0, 0, 4), (4,4,4,5), (0, 0, 7, 6)} from [3]. Algorithm 4 computes 55 different order ideals for I(X).…”

“…However, he did not provide any algorithm to find all such bases. Later, in [3], G. Braun and S. Pokutta used polyhedral theory and adapted the classical border basis algorithm to calculate all border bases for an ideal of points.…”

“…However, he did not provide any algorithm to find all such bases. Later, in [3], G. Braun and S. Pokutta used polyhedral theory and adapted the classical border basis algorithm to calculate all border bases for an ideal of points. Let us exhibit an example from [7] which shows that there exists a border basis which cannot be obtained by any algorithm based on a term ordering strategy or the algorithm by G. Braun and S. Pokutta.…”

On the Weak Order Ideal Associated to Linear Codes

“…In addition, for any weight compatible order ≺ every strictly decreasing sequence terminates (due to the finiteness of the set F n q ). In the binary case the behavior of the coset leaders can be translated to the fact that the set of coset leader is an order ideal of F n 2 ; whereas, for non binary linear codes this is no longer true even if we try to use the characterization of order ideals given in [4], where order ideals do not need to be associated with admissible orders. Definition 6.…”