Can an ideal I in a polynomial ring $$\Bbbk [\mathbf {x}]$$ k [ x ] over a field be moved by a change of coordinates into a position where it is generated by binomials $$\mathbf {x}^\mathbb A- \lambda \mathbf {x}^\mathbf {b}$$ x A - λ x b with $$\lambda \in \Bbbk $$ λ ∈ k , or by unital binomials (i.e., with $$\lambda = 0$$ 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 $$\mathcal {I}$$ 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 $$\mathcal {I}$$ I
is contained in the fiber of a second family $$\mathcal {I}'$$ 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 $$\mathcal {I}$$ 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; or
that admit a faithful multigrading by $$\mathbb {Z}^r$$ 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.