Context-free languages in Algebraic Geometry and Number Theory Jos - PowerPoint PPT Presentation

Context-free languages in Algebraic Geometry and Number Theory Jos´ e Manuel Rodr´ ıguez Caballero Ph.D. student Laboratoire de combinatoire et d’informatique math´ ematique (LaCIM) eal (UQ` Universit´ e du Qu´ ebec ` a Montr´ AM) Presentation at the LFANT Seminar Institut de Math´ ematiques de Bordeaux (IMB) Universit´ e de Bordeaux October 24, 2017 work-in-progress Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Motivation To simultaneously compute several arithmetical functions by means of several language-theoretical computations, but just doing one number-theoretical computation. Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Part I Kassel-Reutenauer polynomials Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Computational definition of P n ( q ) Definition For any integer n ≥ 1, we define the n th Kassel-Reutenauer polynomial as follows : � q n − 1+ k + q n − 1 − k � n − 1 � P n ( q ) := a n , 0 + a n , k , k =1 � � d − 2 n d ≤ 2 k < 2 d − n where a n , k := # d | n : . Furthermore, d we define C n ( q ) := ( q − 1) 2 P n ( q ). Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Generating function Theorem ∞ ∞ � (1 − t m ) 2 � C n ( q ) t n . (1 − qt m ) (1 − q − 1 t m ) = 1 + q n m =1 n =1 This result is essentially identity (9.2) in Nathan Jacob Fine, “Basic hypergeometric series and applications”, No. 27, American Mathematical Soc., 1988. Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Complex geometric interpretation of C n ( q ) C := Hilb n �� � � �� Consider the Hilbert scheme H n A 1 A 1 C \{ 0 } × C \{ 0 } of n points on the bidimensional complex torus. Theorem (Hausel, Letellier and Rodriguez-Villegas, 2011) For each n ≥ 1 ,the E-polynomial of H n C is C n ( q ) . Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Complex geometric interpretation of P n ( q ) Notice that ( C \{ 0 } ) × ( C \{ 0 } ) acts naturally on � � � � A 1 A 1 C \{ 0 } × C \{ 0 } . This action induces an action of ( C \{ 0 } ) × ( C \{ 0 } ) on H n C . Define the geometric quotient � H n C := H n C // (( C \{ 0 } ) × ( C \{ 0 } )). Theorem (Hausel, Letellier and Rodriguez-Villegas, 2011) For each n ≥ 1 ,the E-polynomial of � H n C is P n ( q ) . Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Evaluation at roots of unity Theorem (Kassel and Reutenauer, 2016) For any integer n ≥ 1 , (i) P n (1) is the sum of divisors of n, (ii) C n ( − 1) is the number of integer solutions to the equation x 2 + y 2 = n, � �� � √− 1 � C n � is the number of integer solutions to the equation (iii) x 2 + 2 y 2 = n, � � − 1+ √− 3 (iv) 6 Re P n is the number of integer solutions to the 2 equation x 2 + xy + y 2 = n. Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Finite fields geometric interpretation of C n ( q ) Consider the Hilbert scheme F q := Hilb n �� � � �� H n A 1 A 1 F q \{ 0 } × F q \{ 0 } of n points on the bidimensional F q -torus. Theorem (Kassel and Reutenauer, 2015) For each n ≥ 1 , t d ∞ dt Z H n F q ( t ) � C n ( q m ) t m = , Z H n F q ( t ) m =1 F q ( t ) is the local zeta function of H n where Z H n F q . Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Evaluation at prime powers Theorem (Kassel and Reutenauer, 2015) For any prime power q and any integer n ≥ 1 , C n ( q ) is the number of ideals I of the group algebra F q [ Z ⊕ Z ] such that F q [ Z ⊕ Z ] / I is a vector space of dimension n over F q . Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

The coefficients of P n ( q ) Theorem (J. M. R. C., 2017) For any integer n ≥ 1 , (i) the largest coefficient of P n ( q ) is F ( n ) , where F ( n ) is the os-Nicolas function a , i.e. Erd¨ F ( n ) := max t > 0 # { d | n : t < d ≤ 2 t } . (ii) the polynomial P n ( q ) has a coefficient greater than 1 if and only if 2 n is the perimeter b of a Pythagorean triangle, (iii) all the coefficients of P n ( q ) are non-zero if and only if n is 2 -densely divisible c . a . Paul Erd¨ os, Jean-Louis Nicolas. M´ ethodes probabilistes et combinatoires en th´ eorie des nombres. Bull. SC. Math 2 (1976) : 301–320. b . The perimeter of a Pythagorean triangle is always an even integer. c . i.e. the quotient of two consecutive divisors of n is less than or equal to 2. Densely divisible numbers were introduced by the international team polymath8 , led by Terence Tao, in order to improve Zhang’s bounded gaps between primes. Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Odd-trapezoidal numbers An integer n ≥ 1 is an odd-trapezoidal number if for each pair of integers a ≥ 1 and k ≥ 1, the equality n = a + ( a + 1) + ( a + 2) + ... + ( a + k − 1) implies that k is odd. Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Odd-trapezoidal numbers Theorem (J. M. R. C., 2017) For any integer n ≥ 1 , we have that n is odd-trapezoidal if and only if a n , 0 ≥ a n , 1 ≥ a n , 2 ≥ ... ≥ a n , n − 1 , where a n , 0 , a n , 1 , a n , 2 , ..., a n , n − 1 are the coefficients in the computational definition of P n ( q ) . Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Conclusion of Part I From the polynomial P n ( q ) it is computationally easy to derive the following information about n : (i) Whether or not 2 n is the perimeter of a Pythagorean triangle. (ii) Whether or not n is 2-densely divisible. (iii) Whether or not n is odd-trapezoidal. (iv) The number of middle divisors 1 of n . (v) The number of integer solutions to the equations x 2 + y 2 = n , x 2 + 2 y 2 = n and x 2 + xy + y 2 = n . (vi) The number of ideals I of the group algebra F q [ Z ⊕ Z ] such that F q [ Z ⊕ Z ] / I is a vector space of dimension n over F q . (vii) The value of Erd¨ os-Nicolas function at n . (viii) The sum of divisors of n . C and � (ix) Topological information about H n H n C . √ 1. i.e the divisors d satisfying � n 2 < d ≤ 2 n . Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Part II Language Theory Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Passage from Part I to Part II (i) We will encode part of the information from Kassel-Reutenauer polynomials into formal words. (ii) We will translate some of the properties satisfied by Kassel-Reutenauer polynomials into language-theoretical statements. Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

The non-zero coefficients of C n ( q ) We can express C n ( q ) ∈ Z [ q ], in a unique way, as follows 2 , C n ( q ) = η 0 q e 0 + η 1 q e 1 + ... + η k q e k , for some η 0 , η 1 , ..., η k ∈ { +1 , − 1 } and e 0 , e 1 , ..., e k ∈ Z satisfying : (i) e 0 ≥ e 1 ≥ ... ≥ e k ≥ 0, (ii) for any 0 ≤ j ≤ k − 1, if e j = e j +1 , then η j = η j +1 . So, the vector KR( n ) := ( η 0 , η 1 , ..., η k ) ∈ { +1 , − 1 } k +1 is well-defined. By abuse of notation, we will write KR( n ) as a word over the alphabet { + , −} , identifying + ↔ +1 and − ↔ − 1. 2. Notice that each positive coefficient of C n ( q ) corresponds to the multipli- city of a pole of Z H n F q ( t ). Similarly, each negative coefficient of C n ( q ) corresponds to the multiplicity of a zero of the same rational function. Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Example For n = 6, q 12 − q 11 + q 7 − 2 q 6 + q 5 − q + 1 C 6 ( q ) = + q 12 − q 11 + q 7 − q 6 − q 6 + q 5 − q + 1 . = Therefore, KR(6) = + − + − − + − + . Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

A hidden pattern Notice that KR(75) = + − − + + − + − − + − + + − + − − + − + + − − + can be obtained from the well-matched parentheses ( ) ( ( ) ) ( ( ) ) ( ) by means of the letter-by-letter substitution 3 µ : { ( , ) } ∗ { + , −} ∗ , − → ( �→ + − , ) �→ − + . KR(75) = + − − + + − + − − + − + + − + − − + − + + − − + ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ( ) ( ( ) ) ( ( ) ) ( ) 3. i.e. morphism of free monoids. Here, Σ ∗ denotes the free monoid over the alphabet Σ. Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT

Well-matched parentheses It follows from the computational definition of C n ( q ) that KR( n ) is palindromic. The following property is less obvious. Theorem (J. M. R. C., 2017) For each integer n ≥ 1 , KR( n ) = µ ( w n ) , for some well-matched parentheses w n . → { ( , ) } ∗ by δ ( n ) := w n . Define the function δ : Z ≥ 1 − Jos´ e Manuel Rodr´ ıguez Caballero Context-free languages in AG and NT