Plane Trees and Algebraic Numbers Brief review of the paper of A. Zvonkin and G. Shabat

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

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

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

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

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

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

8. The bond between plane treesand 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 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 TP TQ TR JASS07 selection talk

10. Thank you Please, any questions JASS07 selection talk

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

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