3 years ago

When is a Polynomial Ideal Binomial After an Ambient Automorphism?

Lukas Katthän, Mateusz Michałek, Ezra Miller

Abstract

Can an ideal I in a polynomial ring \(\Bbbk [\mathbf {x}]\) over a field be moved by a change of coordinates into a position where it is generated by binomials \(\mathbf {x}^\mathbb A- \lambda \mathbf {x}^\mathbf {b}\) with \(\lambda \in \Bbbk \) , 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  \(\mathcal {I}\) 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  \(\mathcal {I}\)

  • is contained in the fiber of a second family  \(\mathcal {I}'\) 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  \(\mathcal {I}\) 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  \(\mathbb {Z}^r\) 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.

  • -Abstract Truncated-

Open access
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