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A sharp bound on the number of real intersection points of a sparse plane curve with a line

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Frédéric Bihan, Boulos El Hilany

Algebraic Geometry · Submitted 10 Jun 2015

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

The authors' main claim We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t -7.

how it is stated · asserted 1.00 read the record →

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arXiv ID1506.03309 (version 1)
Pith IDSXAEFLIT, Pith's permanent record of this paper and its review
CitationBoulos El Hilany, Fr\'ed\'eric Bihan. "A sharp bound on the number of real intersection points of a sparse plane curve with a line." arXiv:1506.03309 (2015). https://pith.science/paper/SXAEFLIT
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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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,904 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t -7. This improves a result by M. Avendano. Furthermore, we prove that this bound is sharp for t = 3 with the help of Grothendieck's dessins d'enfant.

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 We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t -7. result 0.98 0.96 0.96 asserted 1.00
s1 This improves a result by M. Avendano. contribution 0.37 0.37 0.04 asserted 1.00
s2 Furthermore, we prove that this bound is sharp for t = 3 with the help of Grothendieck's dessins d'enfant. result 0.89 0.95 0.11 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

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

We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t -7. This improves a result by M. Avendano. Furthermore, we prove that this bound is sharp for t = 3 with the help of Grothendieck's dessins d'enfant.

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