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Markov Numbers, Christo El Words, and the Uniqueness Conjecture

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Markov Numbers, Christo El Words, and the Uniqueness Conjecture Markov numbers, Christoel words, and the uniqueness conjecture Aldo de Luca Alessandro De Luca Università di Napoli Federico II Giornate di informatica teorica in memoria di Aldo de Luca Roma, 12 luglio 2019 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Outline 1 Characteristic matrices Basics General properties of µ¹wº 2 Frobenius’ uniqueness conjecture The map S : w 7! µ¹wº1;2 Tight bounds and uniqueness The Fibonacci and Pell cases Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 2 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Trace Equals 3 Times Upper Right Denition A matrix M 2 SL2¹Úº is characteristic if tr M = 3M1;2: Example 17 10 M = is characteristic, as 22 13 det M = 17 · 13 − 22 · 10 = 1 and 17 + 13 = 3 · 10. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 3 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Simple Constraints Proposition Let α m M = β γ be characteristic. Elements on the same row or column are coprime, and α 2 ≡ γ2 ≡ −1 ¹mod mº: Also, up to switching α and γ, M is determined by any two elements. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 4 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Markov Triples = Characteristic Products Products of char. matrices need not be characteristic, but: Theorem Let M 0; M 00 be characteristic and M = M 0M 00. Then M is characteristic () the upper right elements m0; m00; m (of M 0; M 00; M respectively) verify the Markov equation, i.e., ¹m0º2 + ¹m00º2 + m2 = 3m0m00m: Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 5 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References ∗ The Morphism µ : fa; bg ! SL2¹Úº Setting 2 1 5 2 µ¹aº = ; µ¹bº = 1 1 2 1 ∗ denes an injective morphism µ : fa; bg ! SL2¹Úº. Note that µ¹aº and µ¹bº are characteristic. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 6 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Reversal and µ Let we denote the reversal of w . For instance, if w = aabab, then we = babaa. Lemma ∗ T For all w 2 fa; bg , µ¹weº = µ¹wº . So, w is a palindrome () µ¹wº is symmetric. Let PAL denote the set of palindromes over fa; bg. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 7 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References More Relations on elements Proposition p q Let w 2 fa; bg+, and let µ¹wº = . Then p > q; r ≥ s. r s ∗ Moreover, q < r () w = uavbue for suitable u; v 2 fa; bg . Proposition p q Let u 2 PAL and µ¹uº = . Then q s q + s ≤ p ≤ 2q + s with p = q + s , u 2 a∗ and p = 2q + s , u 2 b∗. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 8 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Characterizing Characteristic µ¹wº Matrices µ¹wº need not be characteristic; for instance, 5 3 µ¹aaº = is not. 3 2 Theorem Let w 2 fa; bg∗. Then µ¹wº is characteristic () w 2 fa; bg [ aPALb: Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 9 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References A Meaningful Decomposition ∗ Let ν : fa; bg ! SL2¹Úº be dened by 1 0 1 1 ν¹aº = ; ν¹bº = ν¹aºT = : 1 1 0 1 It is injective and a well-known tool in the study of Christoel pairs. It is easy to see that µ = ν ◦ ζ where ζ is the injective endomorphism dened by ζ¹aº = ba; ζ¹bº = bbaa: Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 10 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References A Consequence Recall the palindromization map dened by ¹"º = " and ¹v xº = ¹ ¹vºxº¹+º for v 2 fa; bg∗; x 2 fa; bg where w ¹+º is the right palindromic closure of w . Proposition Let w = aub with u 2 PAL. Then µ¹wº1;2 = ja ¹aζ¹uºbºbj i.e., it is the length of a Christoel word whose directive word aζ¹uºb has an antipalindromic middle ζ¹uº. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 11 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References A Similar, Nicer Point of View The following independent result uses almost the same decomposition: Theorem (Reutenauer & Vuillon 2017) For all v 2 fa; bg∗, µ¹a ¹vºbº1;2 = ja ¹ E ¹avººbj; where E = θ ◦ is the antipalindromization map, and θ is the Thue-Morse morphism (θ¹aº = ab; θ¹bº = ba). Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 12 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Denition of S ∗ Dene a map S : fa; bg ! Î by S¹wº = µ¹wº1;2. 0 Since S¹wº = 1 0 µ¹wº , when viewed as a formal series, 1 Õ S = S¹wºw w 2fa;bg∗ is rational. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 13 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References S as a Series 2a 2b a a 1 2 a 5b 2b b S = ¹2a + 5b + ¹a + 2bº¹a + bº∗¹a + 2bºº∗ ¹a + 2bº¹a + bº∗ Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 14 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Not Injective in General Proposition For all u 2 fa; bg∗, S¹aubº = S¹aube º, S¹aθ¹uºbº = S¹aθ¹ueºbº, where θ is the Thue-Morse morphism. Example S¹aabbº = 75 = S¹ababº; S¹aabbabº = 1130 = S¹abaabbº. Thus, even the restriction of S to fa; bg [ aPALb is not injective. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 15 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References What about Christoel? Let CH be the set of (lower) Christoel words over fa; bg. Theorem (Borel, Laubie 1993, etc.) The following holds: CH = fa; bg [ aPALb \ ¹fa; bg [ aPALbº2 : That is, Christoel words can be recursively dened: w 2 CH () w is either a letter or a word of aPALb that is the concatenation of two shorter Christoel words. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 16 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Hence, Markov Triples We have seen that: 1 w 2 CH () w 2 fa; bg or w 2 aPALb and w = w 0w 00 with w 0; w 00 2 fa; bg [ aPALb; 2 w 2 fa; bg [ aPALb () µ¹wº is characteristic; 0 00 0 00 3 Characteristic matrices M ; M ; M M correspond (by their upper right elements) to Markov triples. Corollary (see Cohn 1972, Reutenauer 2009, etc.) S maps Christoel pairs to (nonsingular) Markov triples, i.e., if w = w 0w 00, w; w 0; w 00 2 CH, then S¹w 0º2 + S¹w 00º2 + S¹wº2 = 3S¹w 0ºS¹w 00ºS¹wº: Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 17 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References The Conjecture: S Injective on Christoel The previous map is actually a bijection, so that the Conjecture (Frobenius 1913) Markov triples are uniquely determined by their maximal element. is equivalent to Conjecture The restriction S jCH is injective. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 18 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Evidence Suggesting More Conjectures Our limited experiments (Markov numbers grow fast!.. .) also suggest that: 0 0 0 ∗ 1 if w , w and S¹wº = S¹w º, then w; w 2 afa; bg b; 2 if w 2 aPALb and S¹wº is a Markov number, then w 2 CH. Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 19 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Proving Uniqueness via Matrices A Markov number m is unique if it is the maximal element of a unique Markov triple, or equivalently, if the set −1 S jCH ¹mº = fw 2 CH j S¹wº = mg is a singleton. Since µ is injective, and matrices µ¹wº are uniquely determined by any two elements, α m m is unique () \! γ : 2 µ¹CHº for suitable α; β : β γ Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 20 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References (Extended) Known Bounds Hence, as γ2 ≡ −1 ¹mod mº, m is closer to uniqueness when this has few solutions for γ. Theorem α m If w 2 aPALb and µ¹wº = , then β γ p p 5 − 1 2 − 2 m < γ < m: 2 $p % l p m 5 − 1 2 − 2 m ≤ γ ≤ m : 2 Aldo de Luca, ADL (Napoli Federico II) Christoel words & the uniquess conj. Rome, July 12, 2019 21 / 30 Characteristic matrices Frobenius’ uniqueness conjecture Conclusions References Tight Versions The previous bounds are tight, since for example l p m w 2 ab∗ =) γ = 2 − 2 m ; p ∗ j 5−1 k w 2 a b =) γ = 2 m .
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