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Algebraic Degrees of Generalized Nash Equilibrium Problems

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Jiawang Nie, Kristian Ranestad, Xindong Tang

Optimization and Control · Submitted 31 Jul 2022

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

The authors' main claim We also give formulae for algebraic degrees of GNEPs, which count the numbers of complex Fritz-John points for generic cases.

how it is stated · asserted 1.00 read the record →

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arXiv ID2208.00357 (version 2)
Pith IDSEAEFAAT, Pith's permanent record of this paper and its review
CitationJiawang Nie, Kristian Ranestad, Xindong Tang. "Algebraic Degrees of Generalized Nash Equilibrium Problems." arXiv:2208.00357 (2022). https://pith.science/paper/SEAEFAAT
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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, 4,274 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.65 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.33 · compares with prior results 0.04 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

This paper studies the algebraic degree of generalized Nash equilibrium problems (GNEPs) given by polynomials. Their generalized Nash equilibria (GNEs), as well as their KKT or Fritz-John points, are algebraic functions in the coefficients of defining polynomials. We study the degrees of these algebraic functions, which also count the numbers of complex KKT or Fritz-John points. Under some genericity assumptions, we show that a GNEP has only finitely many complex Fritz-John points and every Fritz-John point is a KKT point. We also give formulae for algebraic degrees of GNEPs, which count the numbers of complex Fritz-John points for generic cases.

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 studies the algebraic degree of generalized Nash equilibrium problems (GNEPs) given by polynomials. contribution 0.82 0.83 0.07 asserted 0.93
s1 Their generalized Nash equilibria (GNEs), as well as their KKT or Fritz-John points, are algebraic functions in the coefficients of defining polynomials. result 0.90 0.42 0.23 asserted 1.00
s2 We study the degrees of these algebraic functions, which also count the numbers of complex KKT or Fritz-John points. contribution 0.90 0.91 0.21 asserted 0.99
s3 Under some genericity assumptions, we show that a GNEP has only finitely many complex Fritz-John points and every Fritz-John point is a KKT point. result 0.99 0.94 0.23 asserted 1.00
s4 We also give formulae for algebraic degrees of GNEPs, which count the numbers of complex Fritz-John points for generic cases. contribution 0.64 0.95 0.41 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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  • 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

This paper studies the algebraic degree of generalized Nash equilibrium problems (GNEPs) given by polynomials. Their generalized Nash equilibria (GNEs), as well as their KKT or Fritz-John points, are algebraic functions in the coefficients of defining polynomials. We study the degrees of these algebraic functions, which also count the numbers of complex KKT or Fritz-John points. Under some genericity assumptions, we show that a GNEP has only finitely many complex Fritz-John points and every Fritz-John point is a KKT point. We also give formulae for algebraic degrees of GNEPs, which count the numbers of complex Fritz-John points for generic cases.

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3 arXiv papers cite this in 19 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 · 13 sentences without a confident label

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Cites it as support 3 papers

What it cites · 4 sentences without a confident label

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