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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; 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; 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 numbers Definition and properties Theorem (Markov) All Markov triples lie on the Markov tree. The entries of all Markov triples form the sequence of Markov numbers 1; 2; 5; 13; 29; 34; 89; 169;::: (A002559 in the OEIS) Every Markov number mn is the maximum of some Markov triple (x; y; mn). Conjecture (Frobenius) Given a Markov number m, there is exactly one Markov triple of the form (x; y; m). Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov numbers Subsequence All odd-indexed Fibonacci number is a Markov number: 0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233::: (A000045) ··· (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 numbers Subsequence All odd-indexed Pell number is a Markov number: 0; 1; 2; 5; 12; 29; 70; 169; 408; 985; ::: (A000129) ··· (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 Related work The equation x2 + y2 + z2 = axyz has positive integral solutions if and only if a = 1; 3. The positive integral solutions of 2 2 x1 + ··· + xn = ax1 ··· xn are also “finitely generated”. Over orders in a number field, the set of solutions 2 2 x1 + ··· + xn = ax1 ··· xn may or may not be “finitely generated”. See work by Silverman and Baragar. Several authors have investigated the solutions of Markov equation and its variations over finite fields. Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov equation over polynomial rings Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Rational numbers vs Function Fields Close your eyes and think of the few slides from Matilde Lalín’s and Allysa Lumley’s talks. Goal: Find all “positive” solutions to the Markov equation equation over the polynomial ring K[t], where K is a field. Every non-zero element of K[t] can be written as αf(t) where f(t) is monic and α 2 K∗. ∗ (Every element n 2 Z satisfies n = (±1)~n, with n~ > 0.) Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov equation over polynomials Monic solutions Let K be a field of characteristic 6= 2. For A monic, x2 + y2 + z2 = Axyz has a monic solution if and only if A = 1. x2 + y2 + z2 = xyz p has a monic solution if and only if i = −1 2 K. If β = 2i then (t; t + β; t2 + βt − 2) is a monic solution. Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov equation New solutions from old solutions x2 + y2 + z2 = xyz (x1; x2; x3) −! (xπ(1); xπ(2); xπ(3)) (zy − x; y; z) (x; y; z) (x; xz − y; z) (x; y; xy − z) (x; y; z) −! (x(f); y(f); z(f)), where f is any monic polynomial. Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov tree of monic solutions (zy − x; y; z) (x; y; xy − z) (x; y; z) (x; xz − y; z) ··· (t; t3 + βt2 − 3t − β; t4 + βt3 − 4t2 − 2βt + 2) ··· (t; t2 + βt − 2; t3 + βt2 − 3t − β) ··· (t2 + βt − 2; t3 + βt2 − 3t − β; t5 + 2βt4 − 9t3 − 6βt2 + 9t + 2β) ··· (t; t + β; t2 + βt − 2) ··· (t2 + βt − 2; t3 + 2βt2 − 7t − 2β; t5 + 3βt4 − 17t3 − 13βt2 + 21t + 3β) ··· (t + β; t2 + βt − 2; t3 + 2βt2 − 7t − 2β) ··· (t + β; t3 + 2βt2 − 7t − 2β; t4 + 3βt3 − 16t2 − 10βt + 10) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov tree of monic solutions (zy − x; y; z) (x; y; xy − z) (x; y; z) (x; xz − y; z) ··· (t; t3 + βt2 − 3t − β; t4 + βt3 − 4t2 − 2βt + 2) ··· (t; t2 + βt − 2; t3 + βt2 − 3t − β) ··· (t2 + βt − 2; t3 + βt2 − 3t − β; t5 + 2βt4 − 9t3 − 6βt2 + 9t + 2β) ··· (t; t + β; t2 + βt − 2) ··· (t2 + βt − 2; t3 + 2βt2 − 7t − 2β; t5 + 3βt4 − 17t3 − 13βt2 + 21t + 3β) ··· (t + β; t2 + βt − 2; t3 + 2βt2 − 7t − 2β) ··· (t + β; t3 + 2βt2 − 7t − 2β; t4 + 3βt3 − 16t2 − 10βt + 10) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov tree of monic solutions (zy − x; y; z) (x; y; xy − z) (x; y; z) (x; xz − y; z) ··· (t; t3 + βt2 − 3t − β; t4 + βt3 − 4t2 − 2βt + 2) ··· (t; t2 + βt − 2; t3 + βt2 − 3t − β) ··· (t2 + βt − 2; t3 + βt2 − 3t − β; t5 + 2βt4 − 9t3 − 6βt2 + 9t + 2β) ··· (t; t + β; t2 + βt − 2) ··· (t2 + βt − 2; t3 + 2βt2 − 7t − 2β; t5 + 3βt4 − 17t3 − 13βt2 + 21t + 3β) ··· (t + β; t2 + βt − 2; t3 + 2βt2 − 7t − 2β) ··· (t + β; t3 + 2βt2 − 7t − 2β; t4 + 3βt3 − 16t2 − 10βt + 10) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov tree of monic solutions (zy − x; y; z) (x; y; xy − z) (x; y; z) (x; xz − y; z) ··· (t; t3 + βt2 − 3t − β; t4 + βt3 − 4t2 − 2βt + 2) ··· (t; t2 + βt − 2; t3 + βt2 − 3t − β) ··· (t2 + βt − 2; t3 + βt2 − 3t − β; t5 + 2βt4 − 9t3 − 6βt2 + 9t + 2β) ··· (t; t + β; t2 + βt − 2) ··· (t2 + βt − 2; t3 + 2βt2 − 7t − 2β; t5 + 3βt4 − 17t3 − 13βt2 + 21t + 3β) ··· (t + β; t2 + βt − 2; t3 + 2βt2 − 7t − 2β) ··· (t + β; t3 + 2βt2 − 7t − 2β; t4 + 3βt3 − 16t2 − 10βt + 10) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings (x; y; z) −! (x(t2 + 1); y(t2 + 1); z(t2 + 1)) (t2 + 1 + β; t4 + (β + 2) t2 + β − 1; t6 + (β + 3) t4 + 2βt2 − 2) (t2 + 1; t2 + 1; t4 + (β + 2) t2 + β − 1) (t2 + 1; t4 + (β + 2) t2 + β − 1; t6 + (2β + 3) t4 + (4β − 4) t2 − 6) Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Monic solutions of the Markov equation Theorem (C., Kelly, VanFossen) Up to permutations, all monic solutions of x2 + y2 + z2 = xyz over K lie on a tree with root (f; f + β; f 2 + βf − 2); where f is a monic polynomial and β = 2i. Remark Up to permutation and composition, the monic polynomial solutions of the Markov equation “branch out” of the solution (t; t + β; t2 + βt − 2) Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Monic solutions of the Markov equation Theorem (C., Kelly, VanFossen) Up to permutations, all monic polynomial solutions of x2 + y2 + z2 = xyz lie on a tree with root (f; f + β; f 2 + βf − 2); where f is a monic polynomial and β = 2i.
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