The free Markoff field
نویسندگان
چکیده
منابع مشابه
Combinatorial Interpretations for the Markoff Numbers
In this paper we generalize the concept of Markoff numbers with the construction of Markoff polynomials. We then present a combinatorial model based on a family of recursively-defined graphs which we call snakes, such that the Markoff polynomials and Markoff numbers are respectively weighted and unweighted enumerators of the perfect matchings of snakes. We have found four proofs of the central ...
متن کاملThe Exponent for the Markoff-hurwitz Equations
In this paper, we study the Markoff-Hurwitz equations x0 + ...+x 2 n = ax0 · · ·xn. The variety V defined by this equation admits a group of automorphisms A ∼= Z/2∗· · ·∗Z/2 (an n+1 fold free product). For a solution P on this variety, we consider the number NP (t) of points Q in the A-orbit of P with logarithmic height h(Q) less than t. We show that if a is rational, and P is a non-trivial rat...
متن کاملOn the Uniqueness Conjecture for Markoff Triples
ν(θ) = inf{c : |θ − p/q| < c/q for infinitely many reduced fractions p/q}. The set of values {νi} of the Markoff function in the range ν(θ) > 1/3 is called the Markoff spectrum [7], which is a denumerable infinite set and νi ↓ 1/3. An other definition of the Markoff spectrum uses the integer solutions of the Diophantine equation x + y + z = 3xyz. (1) A solution (x, y, z) to this equation with 0...
متن کاملThe Markoff-Duffin-Schaeffer inequalities abstracted.
The kth Markoff-Duffin-Schaeffer inequality provides a bound for the maximum, over the interval -1 </= x </= 1, of the kth derivative of a normalized polynomial of degree n. The bound is the corresponding maximum of the Chebyshev polynomial of degree n, T = cos(n cos(-1)x). The requisite normalization is over the values of the polynomial at the n + 1 points where T achieves its extremal values....
متن کاملMarkoff numbers and ambiguous classes
The Markoff conjecture states that given a positive integer c, there is at most one triple (a, b, c) of positive integers with a ≤ b ≤ c that satisfies the equation a2 + b2 + c2 = 3abc. The conjecture is known to be true when c is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant d = 9c2 − 4,...
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ژورنال
عنوان ژورنال: Journal of Functional Analysis
سال: 1973
ISSN: 0022-1236
DOI: 10.1016/0022-1236(73)90025-6