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Computing Node Polynomials for Plane Curves

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Florian Block

Algebraic Geometry · Submitted 1 Jun 2010

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

The authors' main claim Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14.

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arXiv ID1006.0218 (version 3)
Pith ID3WCMQ3V5, Pith's permanent record of this paper and its review
CitationFlorian Block. "Computing Node Polynomials for Plane Curves." arXiv:1006.0218 (2010). https://pith.science/paper/3WCMQ3V5
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,400 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.13 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.93 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

According to the G\"ottsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. These "node polynomials" N_delta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14. Furthermore, we improve the threshold of polynomiality and verify G\"ottsche's conjecture on the optimal threshold up to delta <= 14. We also determine the first 9 coefficients of N_delta(d), for general delta, settling and extending a 1994 conjecture of Di Francesco and Itzykson.

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 According to the G\"ottsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. background 1.00 0.03 0.09 asserted 1.00
s1 These "node polynomials" N_delta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively. background 0.98 0.03 0.05 asserted 1.00
s2 Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14. contribution 0.85 0.97 0.52 asserted 1.00
s3 Furthermore, we improve the threshold of polynomiality and verify G\"ottsche's conjecture on the optimal threshold up to delta <= 14. result 0.71 0.95 0.24 asserted 1.00
s4 We also determine the first 9 coefficients of N_delta(d), for general delta, settling and extending a 1994 conjecture of Di Francesco and Itzykson. result 0.75 0.96 0.24 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.89 · evidence from simulation or computation 0.66.

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  • T body sentences
  • C citations
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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

According to the G\"ottsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. These "node polynomials" N_delta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14. Furthermore, we improve the threshold of polynomiality and verify G\"ottsche's conjecture on the optimal threshold up to delta <= 14. We also determine the first 9 coefficients of N_delta(d), for general delta, settling and extending a 1994 conjecture of Di Francesco and Itzykson.

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2 arXiv papers cite this in 2 sentences

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

Papers citing it · 1 sentence without a confident label

Uses its method or tool 1 paper
  • Using the refined multiplicity, in [ref] the $N^L,δ_(Σ)$ are studied using methods similar to those employed in [ref] for the nonrefined Severi degrees.

    Refined curve counting on complex surfaces · Refined, real, and tropical · 0.89

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