On the density of S-adic integers near some projective G-varieties

Youssef Lazar

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On the density of S-adic integers near some projective G-varieties

Youssef Lazar

Number Theory · Submitted 24 Apr 2019

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

The authors' main claim We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a S-adic field has nontrivial S-integral solutions.

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We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a S-adic field has nontrivial S-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give two examples of applications for systems involving quadratic and linear forms.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a S-adic field has nontrivial S-integral solutions.	contribution 0.57	0.95	0.87	asserted 0.98
s1	The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers.	method 1.00	0.67	0.04	asserted 1.00
s2	We give two examples of applications for systems involving quadratic and linear forms.	contribution 0.94	0.94	0.08	asserted 1.00

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Original abstract

We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a S-adic field has nontrivial S-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give two examples of applications for systems involving quadratic and linear forms.

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1 arXiv paper cite this in 1 sentence

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

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Cites it as background 1 paper

In a related direction, Lazar [ref] studied the density of a pair comprising a quadratic and linear form at $S$-integer points, see also the recent paper [ref] .

On the density at integer points of a system comprising an inhomogeneous quadratic form and a linear form · Systems of forms · 0.95

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