Generalized Perron Identity for broken lines
Oleg Karpenkov1 ; Matty van-Son1
1 University of Liverpool Mathematical Sciences Building Liverpool L69 7ZL, United Kingdom
Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 131-144.

Dans cet article, nous généralisons l’identité de Perron pour les minima de Markov. Nous exprimons les valeurs des formes quadratiques binaires à discriminant positif en termes des fractions continues associées aux lignes brisées passant par les points où les valeurs sont calculées.

In this paper, we generalize the Perron Identity for Markov minima. We express the values of binary quadratic forms with positive discriminant in terms of continued fractions associated to broken lines passing through the points where the values are computed.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1071
Classification : 11J06, 11H55
Mots clés : Geometry of continued fractions, Perron Identity, binary quadratic indefinite form
Oleg Karpenkov 1 ; Matty van-Son 1

1 University of Liverpool Mathematical Sciences Building Liverpool L69 7ZL, United Kingdom
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2019__31_1_131_0,
     author = {Oleg Karpenkov and Matty van-Son},
     title = {Generalized {Perron} {Identity} for broken lines},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {131--144},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     doi = {10.5802/jtnb.1071},
     mrnumber = {3994722},
     zbl = {07246516},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1071/}
}
TY  - JOUR
AU  - Oleg Karpenkov
AU  - Matty van-Son
TI  - Generalized Perron Identity for broken lines
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2019
SP  - 131
EP  - 144
VL  - 31
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1071/
DO  - 10.5802/jtnb.1071
LA  - en
ID  - JTNB_2019__31_1_131_0
ER  - 
%0 Journal Article
%A Oleg Karpenkov
%A Matty van-Son
%T Generalized Perron Identity for broken lines
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 131-144
%V 31
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1071/
%R 10.5802/jtnb.1071
%G en
%F JTNB_2019__31_1_131_0
Oleg Karpenkov; Matty van-Son. Generalized Perron Identity for broken lines. Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 131-144. doi : 10.5802/jtnb.1071. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1071/

[1] Thomas W. Cusick; Mary E. Flahive The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, 30, American Mathematical Society, 1989 | MR | Zbl

[2] Harold Davenport On the product of three homogeneous linear forms. I, Proc. Lond. Math. Soc., Volume 13 (1938), pp. 139-145 | DOI | MR | Zbl

[3] Harold Davenport On the product of three homogeneous linear forms. II, Proc. Lond. Math. Soc., Volume 44 (1938), pp. 412-431 | DOI | MR

[4] Harold Davenport On the product of three homogeneous linear forms. III, Proc. Lond. Math. Soc., Volume 45 (1939), pp. 98-125 | DOI | MR | Zbl

[5] Vladimir V. Fock; Alexander B. Goncharov Dual Teichmüller and lamination spaces, Handbook of Teichmüller theory. Volume I (IRMA Lectures in Mathematics and Theoretical Physics), Volume 11, European Mathematical Society, 2007, pp. 647-684 | DOI | Zbl

[6] Oleg Karpenkov Elementary notions of lattice trigonometry, Math. Scand., Volume 102 (2008) no. 2, pp. 161-205 | DOI | MR | Zbl

[7] Oleg Karpenkov On irrational lattice angles, Funct. Anal. Other Math., Volume 2 (2009) no. 2-4, pp. 221-239 | DOI | MR | Zbl

[8] Oleg Karpenkov On determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators, Math. Notes, Volume 88 (2010) no. 1-2, pp. 28-38 Russian version: Mat. Zametki 88 (2010), n° 1, p. 30–42 | DOI | MR | Zbl

[9] Oleg Karpenkov Continued fractions and the second kepler law, Manuscr. Math., Volume 134 (2011) no. 1-2, pp. 157-169 | DOI | MR | Zbl

[10] Oleg Karpenkov Geometry of Continued Fractions, Algorithms and Computation in Mathematics, 26, Springer, 2013 | MR | Zbl

[11] Svetlana Katok Continued fractions, hyperbolic geometry and quadratic forms, MASS selecta: teaching and learning advanced undergraduate mathematics, American Mathematical Society, 2003, pp. 121-160 | MR | Zbl

[12] John Lewis; Don Zagier Period functions and the Selberg zeta function for the modular group, The mathematical beauty of physics (Saclay, 1996) (Advanced Series in Mathematical Physics), Volume 24, World Scientific, 1996, pp. 83-97 | Zbl

[13] Yuri I. Manin; Matilde Marcolli Continued fractions, modular symbols, and noncommutative geometry, Sel. Math., New Ser., Volume 8 (2002) no. 3, pp. 475-521 | DOI | MR | Zbl

[14] Andreĭ Markov Sur les formes quadratiques binaires indefinies. (second mémoire), Math. Ann., Volume 17 (1880), pp. 379-399

[15] Oskar Perron Über die Approximation irrationaler Zahlen durch rationale II, Heidelberger Akademie der Wissenschaften, 1921 | Zbl

[16] Alfonso Sorrentino; Alexander P. Veselov Markov Numbers, Mather’s β function and stable norm (2017) (https://arxiv.org/abs/1707.03901) | Zbl

Cité par Sources :