Visualizing Dessins D Enfants

Transcription

1 Willamette Valley Consortium for Mathematics Research Occidental College and Ave Maria University MAA MathFest August 7, 2014

2 Introduction Belyi Maps Dessins Passports Shabat Polynomials Question Example General Results

3 Belyi Maps Definition A Belyi Map is a function F : X C with the critical values in the set {0, 1, } X is a Riemann Surface C = C { } is equivalent to a sphere in R 3

4 Definition A Dessin D Enfant, or Dessin for short, is a connected bicolored graph that is embedded on a Riemann surface and is obtained by a Belyi function, f, in the following way: F 1 (0) black vertices F 1 (1) white vertices (shown as red in our pictures) F 1 ([0, 1]) edges

5 Examples of Dessins and Their Belyi Maps Imz z 16 z z z 1 z3 35 z 4 40 z 3 48 z 2 64 z Imz Rez Rez (4z 1) z 1.5 (1 z) 3 (35z 4 +40z 3 +48z 2 +64z+128)

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7 Passports Definition A Passport is a set of valencies (or degrees) that corresponds to the vertices of the dessin, which is represented in the form [b 1, b 2,..., b n ; w 1, w 2,..., w k ] where b 1...b n are the the black degrees and w 1,..., w k are the white degrees. Example [5, 4, 3; 3, 1, 1, 1, 1, 1, 1, 1, 1, 1] can also be written as [5, 4, 3; 3, 1 9 ]

8 Shabat Polynomials A Shabat Polynomial refers to a Belyi Map defined by F : C C, which is represented by a tree (a graph with no cycles). given a tree we can find its Shabat polynomial by solving: F (z) = 0 black vertices F (z) 1 = 0 white vertices

9 Question Can we find all Belyi maps for a given passport of size k? z 4 1 z z3 2 z 2 3 z z 4 28 z z z Im z Re z 1 [3 3, 1 5 ; 2 7 ] 2

10 Example The passport [s 2, r 2 ; 4, 1 (r 1)2+(s 1)2 ] is size k = 2. [4 2, 2 2 ; 4, 1 8 ]

11 F (z) = A(z c 0 ) 4 (z c 1 ) 4 (z c 2 ) 2 (z c 3 ) 2 F (z) 1 = A(z d 0 ) 4 (z 8 + d 8 z 7 + d 7 z d 2 z + d 1 ) Belyi map when s = 4, r = 2: F (z) = (2z2 + 4) 4 (z 2 1) 2 General Form: F (z) = s s ( 1) r (z 2 1) r (s +rz 2 ) s [4 2, 2 2 ; 4, 1 8 ]

12 Belyi Maps for Passports with only one Dessin (k=1 in Shabat-Zvonkin paper) Passport Belyi Map [n; 1 n ] z n [2 n, 1; 2 n 1 + cos ((2n + 1) arccos(z)), 1] 2 [2 n ; 2 n 1, cos (2n arccos(z)) ] ( 2 r 1 ) ( t ) s [s r 1, t; r, 1 (r 1)(s 1)+(t 1) ] (1 z) t z k s k k! k=0 [r, t, 1 r+t 2 ; 2 r+t 1 ] 4S r,t (z)(1 S r,t (z)) [n 2, 1 4n 3 ; 3 2n 1 ] 3 ( ) 3 i S r (z) (1 S r (z)) S r (z) 1 i 3 2 [3 3, 1 5 ; 2 7 ] (z ( 1)z3 2z 2 + 3z + 9 ) 3 ( 8z z z z ) Where S r,t (z) = (1 z) t r 1 ( ) t 1 + j z j and S r (z) = S r,r (z) t 1 j=0

13 Belyi Maps for passports with exactly two Dessins (k=2) Passport [r 2, s 2 ; 4, 1 2r+2s 4 ] [r, t, 1 2r+2t 3) ; 3 r+t 1 ] [r 2, 1 2+3k ; 4 k ] [r 2, 1 3+4k ; 5 k ] [3 2, 1; 2 2, 1 3 ] [3, 2 2 ; 2 2, 1 3 ] [3, 2 2, 1 3 ; 2 5 ] [4 3, 1 8 ; 2 10 ] Belyi Map s s ( 1) r (z 2 1) r (s + rz 2 ) s? 3 ( ) 3 i Sr,t(z) (1 Sr,t(z)) Sr,t(z) 1 i 3 2? 16( 1 + Sr (z))sr (z)(1/2 Sr (z) + Sr (z) 2 ) 4( 1 + Sr (z))sr (z)(( 1 + i) (1 + 2i)Sr (z) + Sr (z) 2 ) C( 1 + Sr (z))sr (z)[i( 2 + 5) ) ( 3i + 2i 5 + 5(5 2 5))Sr (z) + (3i + 5(5 2 5))Sr (z) 2 2iSr (z) 3 ] C( 1 + Sr (z))sr (z)(i( i ) (3i + 2i 5 + 5( ))Sr (z) + ( 3i + 5( ))Sr (z) 2 + 2iSr (z) 3 ) C(11i + 7 8iz)( 1 + z) 3 z 3 C(3i iz)( 1 + z) 3 z 3 Cz 3 (35( ) 21( )z +180z 2 ) 2 Cz 3 ( 35( i ) + 21( i )z + 180z 2 ) (z 1)z3 (8 + 4z + 3z 2 ) 2 ( z + 9z 2 ) (z 1)2 z 2 (3 + 2z) 3 ( 45 10z + 20z 2 + 8z 3 ) 4( 2+z) 4 ( 1+z) 4 z 4 (5 8z +4z 2 )( 1 4z 10z 2 20z 3 +45z 4 24z 5 +4z 6 ) z4 (36 6z + z 2 ) 4 (6 + 4z + z 2 )( z z z z 4 20z 5 + z 6 ) Where C is a large constant (too large for the table!)

14 An Application Let F (z) = (2z2 + 4) 4 (z 2 1) 2 be our equation found earlier. z z z Im z Re z If D(z) = (z d i ) = (3 + z 2 )( 8 + 3z 4 + z 6 ) Then P(z) = 1 8 ( z4 + 8z 6 9z 8 6z 10 z 12 ) and Q(z) = 1 8 z2 ( 1 + z 2 )(2 + z 2 ) 2 is a solution to the equation P(z) 2 D(z)Q(z) 2 = 1

15 The Pell-Abel Equation Question Solve for P(z) 2 D(z)Q(z) 2 = 1 for a given D(z) Theorem Let T (z) be a Shabat Polynomial. Then there exists a corresponding D(z) of the form D(z) = (z d i ), where d i represent the coordinates of the odd vertices, for which P(z) 2 D(z)Q(z) 2 = 1 has infinitely many solutions. Example R(z) = s( s ( 1) r (z 2 1) r (s + rz 2 s where s = 2n and r = 2 n ) ) ) D(z) = kn n k 1 z ( 2n 2k 2 n + n ) kn n k 1 z 2k+2 k=1 ( n+1 k+1 where 2R(z) + 1 and [ 2R(z)+1]2 1 D(z) k=1 ( n+1 k+1 is one of infinitely many solutions.

16 Future Projects and Open Questions Inverse Enumeration for Belyi Maps with multiple faces Belyi Maps for Cartesian Products of graphs Exploration of the role Galois Action plays on Belyi Maps and Dessins Other applications Belyi Maps have on Diophantine Equations

17 Acknowledgements Willamette Valley Consortium for Mathematics Research, NSF Dr. Naiomi Cameron Lewis and Clark College Ave Maria University and Occidental College Dr. Edray Goins from Purdue University Computer Science Whizzes

18 References J. Couveignes, Calcul et rationalité de fonctions de Belyi en genre 0, Annales de l Institut Fourier (Grenoble) 44 (1994), no. 1, L. Granboulan, Calcul d objets géométriques à l aide de méthodes algébraiques et numériques: dessins d enfants, Ph.D. thesis, Université Paris 7, Y. Kochetkov, Geometry of planar trees. (Russian) Fundam. Prikl. Mat. 13 (2007), no. 6, ; translation in J. Math. Sci. (N. Y.) 158 (2009), no. 1, S. Lando, A. Zvonkin, Graphs on surfaces and their applications, Encyclopedia of Mathematical Sciences, Low-Dimensional Topology, II, Springer-Verlag, Berlin, Y. Matiyasevich, Computer evaluation of generalized Chebyshev polynomials, em Moscow Univ. Math. Bull. 51 (1996), no. 6, G. Shabat, A. Zvonkin, Plane trees and algebraic numbers, (English summary) Jerusalem combinatorics 93, , Contemp. Math., 178, Amer. Math. Soc., Providence, RI, L. Schneps, The Grothendieck theory of dessins d enfants (Luminy, 1993), London Math. Soc. Lecture Note Ser., 200, Cambridge Univ. Press, Cambridge, J. Sijsling, J. Voight, On computing Belyi maps, arxiv: v3 [math.nt], 2014.

19 THANK YOU! ANY QUESTIONS?