When is a polynomial ideal binomial after an ambient automorphism?

Lukas Katthaen, Mateusz Michałek, and Ezra Miller

Contact the author: Please use for correspondence this email.
Submission date: 13. Jun. 2017
published in: Foundations of computational mathematics, 19 (2019) 6, p. 1363-1385 
DOI number (of the published article): 10.1007/s10208-018-9405-0
Bibtex
MSC-Numbers: 14Q99, 13P99, 14L30, 13A50, 14M25, 68W30
Link to arXiv: See the arXiv entry of this preprint.

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 −λx^b with λ ∈ k, or by unital binomials (i.e., with 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 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.