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When is a Polynomial Ideal Binomial After an Ambient Automorphism?

Publication ,  Journal Article
Katthän, L; Michałek, M; Miller, E
Published in: Foundations of Computational Mathematics
December 1, 2019

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 xA- λxb 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 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 Iis contained in the fiber of a second family I′ of ideals over B;defines a variety of dimension at least d;is generated by binomials; oris generated by unital binomials. A faster containment algorithm is also presented when the fibers of 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; orthat admit a faithful multigrading by Zr of maximal rank r. Even with no ambient group action given, the final application is an algorithm todecide whether a normal projective variety is abstractly toric. All of these loci in B and subsets of G are constructible.

Duke Scholars

Published In

Foundations of Computational Mathematics

DOI

EISSN

1615-3383

ISSN

1615-3375

Publication Date

December 1, 2019

Volume

19

Issue

6

Start / End Page

1363 / 1385

Related Subject Headings

  • Numerical & Computational Mathematics
  • 49 Mathematical sciences
  • 46 Information and computing sciences
  • 08 Information and Computing Sciences
  • 01 Mathematical Sciences
 

Citation

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Katthän, L., Michałek, M., & Miller, E. (2019). When is a Polynomial Ideal Binomial After an Ambient Automorphism? Foundations of Computational Mathematics, 19(6), 1363–1385. https://doi.org/10.1007/s10208-018-9405-0
Katthän, L., M. Michałek, and E. Miller. “When is a Polynomial Ideal Binomial After an Ambient Automorphism?Foundations of Computational Mathematics 19, no. 6 (December 1, 2019): 1363–85. https://doi.org/10.1007/s10208-018-9405-0.
Katthän L, Michałek M, Miller E. When is a Polynomial Ideal Binomial After an Ambient Automorphism? Foundations of Computational Mathematics. 2019 Dec 1;19(6):1363–85.
Katthän, L., et al. “When is a Polynomial Ideal Binomial After an Ambient Automorphism?Foundations of Computational Mathematics, vol. 19, no. 6, Dec. 2019, pp. 1363–85. Scopus, doi:10.1007/s10208-018-9405-0.
Katthän L, Michałek M, Miller E. When is a Polynomial Ideal Binomial After an Ambient Automorphism? Foundations of Computational Mathematics. 2019 Dec 1;19(6):1363–1385.
Journal cover image

Published In

Foundations of Computational Mathematics

DOI

EISSN

1615-3383

ISSN

1615-3375

Publication Date

December 1, 2019

Volume

19

Issue

6

Start / End Page

1363 / 1385

Related Subject Headings

  • Numerical & Computational Mathematics
  • 49 Mathematical sciences
  • 46 Information and computing sciences
  • 08 Information and Computing Sciences
  • 01 Mathematical Sciences