Solutions of the Markov Equation Over Polynomial Rings
Solutions of the Markov equation over polynomial rings Ricardo Conceição, R. Kelly, S. VanFossen 49th John H. Barrett Memorial Lectures University of Tennessee, Knoxville, TN Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov equation Introduction In his work on Diophantine approximation, Andrei Markov studied the equation x2 + y2 + z2 = 3xyz: The structure of the integral solutions of the Markov equation is very interesting. Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov equation Automorphisms x2 + y2 + z2 = 3xyz (x1; x2; x3) −! (xπ(1); xπ(2); xπ(3)) (x; y; z) −! (−x; −y; z) A Markov triple (x; y; z) is a solution of the Markov equation that satisfies 0 < x ≤ y ≤ z. (x; y; 3xy − z) (x; y; z) (x; 3xz − y; z) (3zy − x; y; z) Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov Tree x2 + y2 + z2 = 3xyz (3zy − x; y; z) (x; y; 3xy − z) (x; y; z) (x; 3xz − y; z) ··· (2,29,169) ··· (2,5,29) ··· (5,29,433) ··· (1,1,1) (1,1,2) (1,2,5) ··· (5,13,194) ··· (1,5,13) ··· (1,13,34) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov Tree x2 + y2 + z2 = 3xyz (3zy − x; y; z) (x; y; 3xy − z) (x; y; z) (x; 3xz − y; z) ··· (2,29,169) ··· (2,5,29) ··· (5,29,433) ··· (1,1,1) (1,1,2) (1,2,5) ··· (5,13,194) ··· (1,5,13) ··· (1,13,34) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov Tree x2 + y2 + z2 = 3xyz (3zy − x;
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