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Affine map equivalence versus critical set equivalence for quadratic maps of the plane

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Chia-Hsing Nien, Bruce B. Peckham, Richard P. McGehee

Dynamical Systems · Submitted 31 Aug 2017

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

The authors' main claim As a consequence, the relationship between affine map equivalence and critical set equivalence is established.

how it is stated · asserted 1.00 read the record →

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arXiv ID1708.09506 (version 1)
Pith IDWRGZKLRP, Pith's permanent record of this paper and its review
CitationBruce B. Peckham, Chia-Hsing Nien, Richard P. McGehee. "Affine map equivalence versus critical set equivalence for quadratic maps of the plane." arXiv:1708.09506 (2017). https://pith.science/paper/WRGZKLRP
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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.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,084 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.15 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

In recent work [Nien et al. 2016], the authors enumerated a classification of quadratic maps of the plane according to their critical sets and images. It is straightforward to show that quadratic maps which are affinely map equivalent are also equivalent in the critical set classification. The question remained whether maps that are equivalent in the critical set classification are also affinely map equivalent. This paper establishes a complete enumeration of the affine map equivalence classes. As a consequence, the relationship between affine map equivalence and critical set equivalence is established. In short, there are eighteen affine map equivalence classes. Three pairs of those classes have critical sets and images that match, but each pair has some other geometric property, preserved by affine map equivalence, that does not match. The other twelve affine map equivalence classes and the critical set equivalences are in one-to-one correspondence.

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 In recent work [Nien et al. 2016], the authors enumerated a classification of quadratic maps of the plane according to their critical sets and images. background 1.00 0.03 0.02 asserted 1.00
s1 It is straightforward to show that quadratic maps which are affinely map equivalent are also equivalent in the critical set classification. background 0.97 0.21 0.04 asserted 1.00
s2 The question remained whether maps that are equivalent in the critical set classification are also affinely map equivalent. gap 1.00 0.05 0.04 asserted 0.44
s3 This paper establishes a complete enumeration of the affine map equivalence classes. contribution 0.95 0.94 0.20 asserted 1.00
s4 As a consequence, the relationship between affine map equivalence and critical set equivalence is established. result 0.66 0.85 0.58 asserted 1.00
s5 In short, there are eighteen affine map equivalence classes. result 0.99 0.90 0.43 asserted 1.00
s6 Three pairs of those classes have critical sets and images that match, but each pair has some other geometric property, preserved by affine map equivalence, that does not match. result 0.95 0.87 0.32 asserted 1.00
s7 The other twelve affine map equivalence classes and the critical set equivalences are in one-to-one correspondence. result 0.97 0.89 0.30 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • 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

In recent work [Nien et al. 2016], the authors enumerated a classification of quadratic maps of the plane according to their critical sets and images. It is straightforward to show that quadratic maps which are affinely map equivalent are also equivalent in the critical set classification. The question remained whether maps that are equivalent in the critical set classification are also affinely map equivalent. This paper establishes a complete enumeration of the affine map equivalence classes. As a consequence, the relationship between affine map equivalence and critical set equivalence is established. In short, there are eighteen affine map equivalence classes. Three pairs of those classes have critical sets and images that match, but each pair has some other geometric property, preserved by affine map equivalence, that does not match. The other twelve affine map equivalence classes and the critical set equivalences are in one-to-one correspondence.

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