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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