MULTIVARIATE NORMAL DISTRIBUTION FOR INTEGRAL POINTS ON VARIETIES

Daniel El-Baz; Daniel Loughran; Efthymios Sofos

doi:10.1090/tran/8545

MULTIVARIATE NORMAL DISTRIBUTION FOR INTEGRAL POINTS ON VARIETIES

Daniel El-Baz, Daniel Loughran, Efthymios Sofos

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

Given a variety with coefficients in Z, we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdos-Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.

Original language	English
Pages (from-to)	3089-3128
Number of pages	40
Journal	Transactions of the American Mathematical Society
Volume	375
Issue number	5
DOIs	https://doi.org/10.1090/tran/8545
Publication status	Published - 2022

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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title = "MULTIVARIATE NORMAL DISTRIBUTION FOR INTEGRAL POINTS ON VARIETIES",

abstract = "Given a variety with coefficients in Z, we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdos-Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.",

author = "Daniel El-Baz and Daniel Loughran and Efthymios Sofos",

note = "Funding Information: Received by the editors September 6, 2020, and, in revised form, May 27, 2021. 2020 Mathematics Subject Classification. Primary 14G05; Secondary 60F05, 11N36. The first author was supported by the Austrian Science Fund (FWF), projects F-5512 and Y-901. The second author was supported by EPSRC grant EP/R021422/2. Publisher Copyright: {\textcopyright} 2022 American Mathematical Society.",

year = "2022",

doi = "10.1090/tran/8545",

language = "English",

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AU - El-Baz, Daniel

AU - Loughran, Daniel

AU - Sofos, Efthymios

N1 - Funding Information: Received by the editors September 6, 2020, and, in revised form, May 27, 2021. 2020 Mathematics Subject Classification. Primary 14G05; Secondary 60F05, 11N36. The first author was supported by the Austrian Science Fund (FWF), projects F-5512 and Y-901. The second author was supported by EPSRC grant EP/R021422/2. Publisher Copyright: © 2022 American Mathematical Society.

PY - 2022

Y1 - 2022

N2 - Given a variety with coefficients in Z, we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdos-Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.

AB - Given a variety with coefficients in Z, we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdos-Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.

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JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

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ER -

MULTIVARIATE NORMAL DISTRIBUTION FOR INTEGRAL POINTS ON VARIETIES

Abstract

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