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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 $x^a - \lambda x^b$ with $\lambda \in k$, or by unital binomials (i.e., with $\lambda = 0$ 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.