When is a Polynomial Ideal Binomial After an Ambient Automorphism?
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 - λx 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 Iis 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; oris 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 Iwhose stabilizer subgroups in T have maximal dimension; orthat admit a faithful multigrading by Z of maximal rank r. Even with no ambient group action given, the final application is an algorithm todecide whether a normal projective variety is abstractly toric. All of these loci in B and subsets of G are constructible. A b ′ r
Katthän, L; Michałek, M; Miller, E
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