Pith. Sign in

RECORD

When is a polynomial ideal binomial after an ambient automorphism?

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Lukas Katthän, Mateusz Micha{l}ek, Ezra Miller

Commutative Algebra · Submitted 12 Jun 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID1706.03629 (version 1)
Pith IDPQDNTJGY, Pith's permanent record of this paper and its review
CitationEzra Miller, Lukas Katth\"an, Mateusz Micha{\l}ek. "When is a polynomial ideal binomial after an ambient automorphism?." arXiv:1706.03629 (2017). https://pith.science/paper/PQDNTJGY
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,130 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.85 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.14 · states quantitative result 0.09 · compares with prior results 0.69 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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 $x^a - cx^b$ with c in k, or by unital binomials (i.e., with c = 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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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 $x^a - cx^b$ with c in k, or by unital binomials (i.e., with c = 0 or 1)? background 0.99 0.07 0.04 unclear 0.77
s1 Can a variety be moved into a position where it is toric? background 0.99 0.06 0.03 unclear 0.70
s2 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. method 0.54 0.55 0.05 asserted 1.00
s3 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. contribution 0.69 0.93 0.83 asserted 1.00
s4 A faster containment algorithm is also presented when the fibers of F are prime. contribution 0.76 0.92 0.06 asserted 1.00
s5 The big-fiber algorithm is probabilistic but likely faster than known deterministic ones. result 0.29 0.73 0.05 hedged possibility 0.96
s6 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. contribution 0.70 0.88 0.05 asserted 1.00
s7 Even with no ambient group action given, the final application is an algorithm to - decide whether a normal projective variety is abstractly toric. contribution 0.83 0.89 0.09 asserted 1.00
s8 All of these loci in B and subsets of G are constructible; in some cases they are closed. result 0.97 0.87 0.12 asserted 0.96

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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 $x^a - cx^b$ with c in k, or by unital binomials (i.e., with c = 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.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.