MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint

When is a polynomial ideal binomial after an ambient automorphism?

Lukas Katthaen, Mateusz Michałek and Ezra Miller


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.

Jun 13, 2017
Jun 28, 2017
MSC Codes:
14Q99, 13P99, 14L30, 13A50, 14M25, 68W30

Related publications

2019 Journal Open Access
Lukas Katthän, Mateusz Michałek and Ezra Miller

When is a polynomial ideal binomial after an ambient automorphism?

In: Foundations of computational mathematics, 19 (2019) 6, pp. 1363-1385