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

Anton Sadovnikov

Saint-Petersburg State University,

Mathematics & Mechanics Dpt.

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The main finding

The world of bicoloured plane trees is as

rich as that of algebraic numbers.

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Generalized Chebyshev polynomials

P is generalized Chebyshev polynomial,

if it has at most 2 critical values.

Examples:• P(z) = zn

• P(z) = Tn(z)

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The inverse image of a segment

P is a generalized Chebyshev polynomial, the ends of segment are the only critical values

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Examples: Star and chain

P(z) = zn

segment: [0,1]

P(z) = Tn(z)

segment: [-1,1]

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The main theorem

{(plane

bicoloured)

trees}

{(classes of

equivalence of)

generalized

Chebyshev

polynomials}

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Canonical geometric form

Every plane tree has a unique canonical

geometric form

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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

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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

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Thank you

Please, any questions

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Critical points and critical values

If P´(z) = 0 then• z is a critical point• w = P(z) is a critical value

ADDENDUM

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Inverse images

The ends of segment are the only critical values

The segment does not include critical values

ADDENDUM