Can an ideal 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 - tx^b
with t in k, or by unital binomials (i.e., with t=0 or t=1)? Can a variety
be moved into a position where it is toric? 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 of I' of ideals over B;
- defines a variety of dimension at least d;
- is generated by binomials; or
- is generated by unital binomials.

This talk is based on work by Lukas Katthän, Mateusz Michalek, and Ezra Miller;
paper available at https://arxiv.org/pdf/1706.03629.pdf.