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Forms of differing degrees over number fields

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Christopher Frei, Manfred Madritsch

Number Theory · Submitted 19 Dec 2014

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

The authors' main claim As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_K^m$ satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree.

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arXiv ID1412.6419 (version 2)
Pith IDIYW72VHJ, Pith's permanent record of this paper and its review
CitationChristopher Frei, Manfred Madritsch. "Forms of differing degrees over number fields." arXiv:1412.6419 (2014). https://pith.science/paper/IYW72VHJ
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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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,969 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.58 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.30 · compares with prior results 0.97 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_K^m$ satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$. Our main tool is Skinner's number field version of the Hardy-Littlewood circle method. As a by-product, we point out and correct an error in Skinner's treatment of the singular integral.

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 Consider a system of polynomials in many variables over the ring of integers of a number field $K$. background 0.96 0.13 0.02 unclear 0.80
s1 We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. result 0.84 0.97 0.38 asserted 1.00
s2 As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_K^m$ satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree. interpretation 0.58 0.82 0.80 asserted 0.97
s3 This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$. background 0.88 0.19 0.04 asserted 0.99
s4 Our main tool is Skinner's number field version of the Hardy-Littlewood circle method. method 1.00 0.32 0.02 asserted 1.00
s5 As a by-product, we point out and correct an error in Skinner's treatment of the singular integral. contribution 0.55 0.94 0.02 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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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • 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

Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_K^m$ satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$. Our main tool is Skinner's number field version of the Hardy-Littlewood circle method. As a by-product, we point out and correct an error in Skinner's treatment of the singular integral.

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