Rational lines on cubic hypersurfaces

Brandes, Julia; Dietmann, Rainer

doi:10.1017/s0305004120000079

Cited by 5 publications

(8 citation statements)

References 27 publications

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“…In the direction of Birch's first theorem, prime solutions to (1.1) has previously been studied by Brüdern et al [6]. In the special case of a cubic form, the number of variables required in [6] has subsequently been reduced by Brandes and Dietmann [4].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Vaccine Coverage and Effectiveness in a School-Based Varicella Outbreak in Jinan Prefecture, Shandong Province

Liu

et al. 2022

Vaccines

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Background: Licheng District of Jinan Prefecture reported a school-based varicella outbreak. We conducted an investigation to analyze the epidemiology and scope of the outbreak, determine varicella vaccine coverage on the school campus, and estimate varicella vaccine effectiveness (VE). Methods: In the epidemiological investigation, we determined the attack rate, the clinical manifestations of varicella cases, and histories of prior varicella disease and varicella vaccination. We tested students for presence of serum IgM antibodies, and we attempted to isolate the varicella virus from vesicular fluid samples. We used chi-square to compare incidences between classes and floors. VE was estimated using a retrospective cohort study. Results: There were 13 varicella cases in the outbreak. All were among fourth grade students - twelve in Class 7 and one in Class 6. The attack rate in the two classrooms was 14.3% (13/91). Clinical symptoms were rash (100%) and fever (46.15%). All cases were reported within one average incubation period, and the epidemic curve suggested common exposure. Six of the 13 cases previously received one dose of varicella vaccine with a median time between vaccination and infection of 9 years; the other seven cases had not been vaccinated. Varicella vaccine coverage with one or more doses was 81.31%; 2-dose coverage was 38.15%. The median age of receipt of dose 1 was 1.18 years, and median age for receiving dose 2 was 5.12 years. One-dose varicella VE was 73.2% (95% confidence interval: 37.0%, 88.6%), and two-dose VE was 100%. Conclusions: Varicella vaccine coverage has been gradually increasing in recent years, as ≥1-dose and 2-dose coverage rates are higher in younger children than older children. High one-dose vaccination coverage limited the outbreak scope and led to the breakthrough cases being mild. Mild cases were difficult to detect in a timely manner. Varicella vaccine was highly effective, with 1-dose VE of 73% nine years after vaccination and 2-dose VE of 100%. We strongly recommended that all school students receive two doses of varicella vaccine.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Vaccine Coverage and Effectiveness in a School-Based Varicella Outbreak in Jinan Prefecture, Shandong Province

Liu

et al. 2022

Vaccines

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“…Over the field of p-adic numbers, to our knowledge, it is not even clear what these generic possibilities are. In fact a cubic polynomial over p-adic fields needs to have a lot of variables (22) to ensure that the cubic surface has at least one line (see [BD20,Theorem 1.3…”

Section: Classical and Probabilistic Enumerative Geometrymentioning

confidence: 99%

Probabilistic enumerative geometry over p-adic numbers: linear spaces on complete intersections

Manssour¹,

Lerario²

2022

Annales Henri Lebesgue

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We compute the expectation of the number of linear spaces on a random complete intersection in p-adic projective space. Here "random" means that the coefficients of the polynomials defining the complete intersections are sampled uniformly from the p-adic integers. We show that as the prime p tends to infinity the expected number of linear spaces on a random complete intersection tends to 1. In the case of the number of lines on a random cubic in three-space and on the intersection of two random quadrics in four-space, we give an explicit formula for this expectation.Résumé. -Nous calculons l'espérance du nombre d'espaces linéaires dans des intersections complètes aléatoires dans l'espace projectif p-adique. Ici "aléatoire" signifie que les coefficients des polynômes définissant les intersections complètes sont tirés uniformément parmi les entiers p-adiques. Nous montrons que lorsque le nombre premier p tend vers l'infini, le nombre moyen d'espaces linéaires dans une intersection complète aléatoire tend vers 1. Dans le cas du nombre de droites sur une cubique aléatoire en trois dimensions, et de l'intersection de deux quadriques aléatoires en quatre dimensions, nous donnons une formule explicite pour cette espérance.

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“…When F is a cubic form, recent work of the author jointly with Dietmann [12] shows that (1.1) has non-trivial rational solutions whenever n ≥ 29, but that there may not be any rational solutions when n = 11 or lower. For more general settings, (1.1) has been investigated in a series of papers by the present author [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

The density of rational lines on hypersurfaces: a bihomogeneous perspective

Brandes

2021

Monatsh Math

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Let F be a non-singular homogeneous polynomial of degree d in n variables. We give an asymptotic formula of the pairs of integer points $$(\mathbf {x}, \mathbf {y})$$ ( x , y ) with $$|\mathbf {x}| \leqslant X$$ | x | ⩽ X and $$|\mathbf {y}| \leqslant Y$$ | y | ⩽ Y which generate a line lying in the hypersurface defined by F, provided that $$n > 2^{d-1}d^4(d+1)(d+2)$$ n > 2 d - 1 d 4 ( d + 1 ) ( d + 2 ) . In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of X and Y.

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Rational lines on cubic hypersurfaces

Cited by 5 publications

References 27 publications

Vaccine Coverage and Effectiveness in a School-Based Varicella Outbreak in Jinan Prefecture, Shandong Province

Vaccine Coverage and Effectiveness in a School-Based Varicella Outbreak in Jinan Prefecture, Shandong Province

Probabilistic enumerative geometry over p-adic numbers: linear spaces on complete intersections

The density of rational lines on hypersurfaces: a bihomogeneous perspective

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