Hyperelliptic curves, continued fractions, and Somos sequences

Alfred J. van der Poorten

doi:10.1214/074921706000000239

Hyperelliptic curves, continued fractions, and Somos sequences

Alfred J. van der Poorten

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution

Abstract

We detail the continued fraction expansion of the square root of a monic polynomials of even degree. We note that each step of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. In the quartic and sextic cases we observe explicitly that the parameters appearing in the continued fraction expansion yield integer sequences defined by bilinear relations instancing sequences of Somos type.

Original language	English
Title of host publication	Dynamics & stochastics
Subtitle of host publication	festschrift in Honour of M.S. Keane
Editors	Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy
Place of Publication	Beachwood, Ohio
Publisher	Institute of Mathematical Statistics
Pages	212-224
Number of pages	13
ISBN (Print)	0940600641
DOIs	https://doi.org/10.1214/074921706000000239
Publication status	Published - 2006
Event	Workshop on Dynamical systems, Probability theory, and Statistical Mehanics - Eindhoven, Netherlands Duration: 3 Jan 2005 → 7 Jan 2005

Publication series

Name	Lecture notes-monograph series
Publisher	Institute of Mathematical Statistics
Volume	48

Workshop

Workshop	Workshop on Dynamical systems, Probability theory, and Statistical Mehanics
City	Eindhoven, Netherlands
Period	3/01/05 → 7/01/05

Keywords

continued fraction expansion
function field of characteristic zero
hyperelliptic curve
Somos sequence

Access to Document

10.1214/074921706000000239

Cite this

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title = "Hyperelliptic curves, continued fractions, and Somos sequences",

abstract = "We detail the continued fraction expansion of the square root of a monic polynomials of even degree. We note that each step of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. In the quartic and sextic cases we observe explicitly that the parameters appearing in the continued fraction expansion yield integer sequences defined by bilinear relations instancing sequences of Somos type.",

keywords = "continued fraction expansion, function field of characteristic zero, hyperelliptic curve, Somos sequence",

author = "{van der Poorten}, {Alfred J.}",

year = "2006",

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language = "English",

isbn = "0940600641",

series = "Lecture notes-monograph series",

publisher = "Institute of Mathematical Statistics",

pages = "212--224",

editor = "Dee Denteneer and {den Hollander}, Frank and Evgeny Verbitskiy",

booktitle = "Dynamics & stochastics",

address = "United States",

note = "Workshop on Dynamical systems, Probability theory, and Statistical Mehanics ; Conference date: 03-01-2005 Through 07-01-2005",

}

van der Poorten, AJ 2006, Hyperelliptic curves, continued fractions, and Somos sequences. in D Denteneer, F den Hollander & E Verbitskiy (eds), Dynamics & stochastics: festschrift in Honour of M.S. Keane. Lecture notes-monograph series, vol. 48, Institute of Mathematical Statistics, Beachwood, Ohio, pp. 212-224, Workshop on Dynamical systems, Probability theory, and Statistical Mehanics, Eindhoven, Netherlands, 3/01/05. https://doi.org/10.1214/074921706000000239

Hyperelliptic curves, continued fractions, and Somos sequences. / van der Poorten, Alfred J.
Dynamics & stochastics: festschrift in Honour of M.S. Keane. ed. / Dee Denteneer; Frank den Hollander; Evgeny Verbitskiy. Beachwood, Ohio: Institute of Mathematical Statistics, 2006. p. 212-224 (Lecture notes-monograph series; Vol. 48).

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution

TY - GEN

T1 - Hyperelliptic curves, continued fractions, and Somos sequences

AU - van der Poorten, Alfred J.

PY - 2006

Y1 - 2006

N2 - We detail the continued fraction expansion of the square root of a monic polynomials of even degree. We note that each step of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. In the quartic and sextic cases we observe explicitly that the parameters appearing in the continued fraction expansion yield integer sequences defined by bilinear relations instancing sequences of Somos type.

AB - We detail the continued fraction expansion of the square root of a monic polynomials of even degree. We note that each step of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. In the quartic and sextic cases we observe explicitly that the parameters appearing in the continued fraction expansion yield integer sequences defined by bilinear relations instancing sequences of Somos type.

KW - continued fraction expansion

KW - function field of characteristic zero

KW - hyperelliptic curve

KW - Somos sequence

U2 - 10.1214/074921706000000239

DO - 10.1214/074921706000000239

M3 - Conference proceeding contribution

SN - 0940600641

T3 - Lecture notes-monograph series

SP - 212

EP - 224

BT - Dynamics & stochastics

A2 - Denteneer, Dee

A2 - den Hollander, Frank

A2 - Verbitskiy, Evgeny

PB - Institute of Mathematical Statistics

CY - Beachwood, Ohio

T2 - Workshop on Dynamical systems, Probability theory, and Statistical Mehanics

Y2 - 3 January 2005 through 7 January 2005

ER -

Hyperelliptic curves, continued fractions, and Somos sequences

Abstract

Publication series

Workshop

Keywords

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