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A sharp version of Shimizu's theorem on entire automorphic functions

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Ronen Peretz

Complex Variables · Submitted 29 Sep 2017

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

The authors' main claim The third result gives the precise form of an automorphic function that is uniform over an open subset of $\mathbb{C}$.

how it is stated · asserted 1.00 read the record →

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arXiv ID1709.10265 (version 1)
Pith IDOUJAFLEE, Pith's permanent record of this paper and its review
CitationRonen Peretz. "A sharp version of Shimizu's theorem on entire automorphic functions." arXiv:1709.10265 (2017). https://pith.science/paper/OUJAFLEE
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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.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,260 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.22 · compares with prior results 0.79 · mentions code or data availability 0.03.

The abstract, sentence by sentence

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

This paper develops further the theory of the automorphic group of non-constant entire functions. This theory essentially started with two remarkable papers of Tatsujir\^o Shimizu that were published in 1931. There are three results in this paper. The first result is that the ${\rm Aut}(f)$-orbit of any complex number has no finite accumulation point. The second result is an accurate computation of the derivative of an automorphic function of an entire function at any of its fixed points. The third result gives the precise form of an automorphic function that is uniform over an open subset of $\mathbb{C}$. This last result is a follow up of a remarkable theorem of Shimizu. It is a sharp form of his result. It leads to an algorithm of computing the entire automorphic functions of entire functions. The complexity is computed using an height estimate of a rational parameter discovered by Shimizu.

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 This paper develops further the theory of the automorphic group of non-constant entire functions. contribution 0.76 0.90 0.05 asserted 1.00
s1 This theory essentially started with two remarkable papers of Tatsujir\^o Shimizu that were published in 1931. background 1.00 0.03 0.02 asserted 1.00
s2 There are three results in this paper. contribution 0.89 0.63 0.03 asserted 1.00
s3 The first result is that the ${\rm Aut}(f)$-orbit of any complex number has no finite accumulation point. result 1.00 0.91 0.19 asserted 1.00
s4 The second result is an accurate computation of the derivative of an automorphic function of an entire function at any of its fixed points. result 1.00 0.93 0.11 asserted 1.00
s5 The third result gives the precise form of an automorphic function that is uniform over an open subset of $\mathbb{C}$. result 1.00 0.94 0.31 asserted 1.00
s6 This last result is a follow up of a remarkable theorem of Shimizu. background 0.62 0.69 0.10 asserted 1.00
s7 It is a sharp form of his result. interpretation 0.67 0.75 0.20 asserted 1.00
s8 It leads to an algorithm of computing the entire automorphic functions of entire functions. interpretation 0.69 0.83 0.17 asserted 1.00
s9 The complexity is computed using an height estimate of a rational parameter discovered by Shimizu. method 0.97 0.72 0.06 asserted 1.00

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

What was examined, and what was not

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  • figures and tables
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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

This paper develops further the theory of the automorphic group of non-constant entire functions. This theory essentially started with two remarkable papers of Tatsujir\^o Shimizu that were published in 1931. There are three results in this paper. The first result is that the ${\rm Aut}(f)$-orbit of any complex number has no finite accumulation point. The second result is an accurate computation of the derivative of an automorphic function of an entire function at any of its fixed points. The third result gives the precise form of an automorphic function that is uniform over an open subset of $\mathbb{C}$. This last result is a follow up of a remarkable theorem of Shimizu. It is a sharp form of his result. It leads to an algorithm of computing the entire automorphic functions of entire functions. The complexity is computed using an height estimate of a rational parameter discovered by Shimizu.

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