1. JASS07 selection talk 1 Plane Trees and Algebraic Numbers Brief review of the paper of A. Zvonkin and G. Shabat Anton Sadovnikov Saint-Petersburg State University, Mathematics & Mechanics Dpt. JASS 2007 selection talk

2. JASS07 selection talk 2 The main finding The world of bicoloured plane trees is as rich as that of algebraic numbers.

3. JASS07 selection talk 3 Generalized Chebyshev polynomials P is generalized Chebyshev polynomial, if it has at most 2 critical values. Examples: P(z) = z n P(z) = T n (z)

4. JASS07 selection talk 4 The inverse image of a segment P is a generalized Chebyshev polynomial, the ends of segment are the only critical values

5. JASS07 selection talk 5 Examples: Star and chain P(z) = z n segment: [0,1] P(z) = T n (z) segment: [-1,1]

6. JASS07 selection talk 6 The main theorem {(plane bicoloured) trees} {(classes of equivalence of) generalized Chebyshev polynomials}

7. JASS07 selection talk 7 Canonical geometric form Every plane tree has a unique canonical geometric form

8. JASS07 selection talk 8 The bond between plane trees and algebraic numbers Г = aut(alg(Q)) – universal Galois group Г acts on alg(Q) Г acts on {P} Г acts on {T} – this action is faithful

9. JASS07 selection talk 9 Composition of trees If P and Q are generalized Chebyshev polynomials and P(0), P(1) lie in {0, 1} then R(z) = P(Q(z)) is also a generalized Chebyshev polynomial TPTP TQTQ TRTR

10. JASS07 selection talk 10 Thank you Please, any questions

11. JASS07 selection talk 11 Critical points and critical values If P´(z) = 0 then z is a critical point w = P(z) is a critical value ADDENDUM

12. JASS07 selection talk 12 Inverse images The ends of segment are the only critical values The segment does not include critical values ADDENDUM