Title: 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 $x^a - cx^b$
with c in k, or by unital binomials (i.e., with c = 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 F 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 F
- is contained in the fiber of a second family F' 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 F 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.