When is a Polynomial Ideal Binomial After an Ambient Automorphism?

Abstract

Can an ideal I in a polynomial ring k[x] over a field be moved by a change of coordinates into a position where it is generated by binomials x a − λx b with λ ∈ k, or by unital binomials (i.e., with λ = 0 or 1)? Can a variety be moved into a position where it is toric? By fibering the G-translates of I over an algebraic group G acting on affine space, these problems are special cases of questions about a family I of ideals over an arbitrary base B. The main results in this general setting are algorithms to find the locus of points in B over which the fiber of I • is contained in the fiber of a second family I ′ of ideals over B;• defines a variety of dimension at least d;• is generated by binomials; or • is generated by unital binomials. A faster containment algorithm is also presented when the fibers of I are prime. The big-fiber algorithm is probabilistic but likely faster than known deterministic ones. Applications include the setting where a second group T acts on affine space, in addition to G, in which case algorithms compute the set of G-translates of I• whose stabilizer subgroups in T have maximal dimension; or • that admit a faithful multigrading by Z r of maximal rank r. Even with no ambient group action given, the final application is an algorithm to• decide whether a normal projective variety is abstractly toric. All of these loci in B and subsets of G are constructible; in some cases they are closed.

Computing the Binomial Part of a Polynomial Ideal

“…The monomial part of a polynomial ideal I, i.e., the ideal generated by all monomials contained in I, is a subideal of Bin(I) and can be computed using homogenization (see Tutorial 50 in [26]). In [21] the authors construct an algorithm for checking whether an ideal is binomial after applying an ambient automorphism. A method for finding sparse polynomials which vanish on an algebraic set is proposed in [16].…”

Toric geometry of path signature varieties

“…However, since we found no general technique, we expect that the answer may be negative. The main obstacle to provide a nontoric example is that a first potential candidate is already too large to be dealt with, either with ad hoc geometric arguments or general computational methods [KMM17].…”