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From Christoffel Words to Markoff Numbers$
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Christophe Reutenauer

Print publication date: 2018

Print ISBN-13: 9780198827542

Published to Oxford Scholarship Online: January 2019

DOI: 10.1093/oso/9780198827542.001.0001

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Markoff’s Theorem for Quadratic Forms

Markoff’s Theorem for Quadratic Forms

(p.63) 9 Markoff’s Theorem for Quadratic Forms
From Christoffel Words to Markoff Numbers

Christophe Reutenauer

Oxford University Press

In this chapter, Markoff’s theorem for quadratic forms is proved. These forms are real, binary, and indefinite. The two first sections are concerned with results which are reminiscent of the work of Gauss: each form is equivalent, under the action of GL2(Z), to a form having a root larger than 1, the other being between − 1 and 0. For such a form, onemay define a bi-infinite chain of forms, using the expansion into continued fractions of both roots; then the infimum of the form is equal to the infimum of the first coefficients of all these forms. In the last section,Markoff’s theoremis deduced: if three times the infimum of a formis larger than the square root of its discriminant, then the formmust be equivalent to aMarkoff form.

Keywords:   indefinite real binary quadratic forms, equivalent form, infimum of a form, Markoff form, inequality

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