
B. Sury (Bangalor). Diophantine equations of the form
f(x)=g(y).
Questions on counting usually involve finding integer
solutions x,y of equations of the form f(x)=g(y) for
polynomials f,g with integer or rational coefficients.
For instance, if a,b,c are rational and ab is nonzero,
then for m>n>2, the Diophantine equation
a C^{m}_{x} + b C^{n}_{y} = c has only finitely many
integer solutions x,y. Another natural example comes from counting
lattice points in generalized octahedra. The number of integral
points on the ndimensional octahedron
x_{1} + x_{2} + ... + x_{n} ≤ r
is given by the polynomial expression
p_{n}(r) = ∑_{i=0}^{n} 2^{i}
C^{i}_{n}C^{i}_{r}
and the question of whether two octahedra of different dimensions
m,n can contain the same number of integral points becomes
equivalent to the
solvability of p_{m}(x)=p_{n}(y) in integers x,y.
It turns out that when m>n>1, the above equation has only finitely many
integral solutions.
The above results and many others appearing in the last decade have
been made possible by a beautiful theorem of Yuri Bilu & Robert
Tichy from 2000. For an absolutely irreducible polynomial
F(x,y)
in ℤ[x,y], a celebrated 1929 theorem due to Siegel shows
the finiteness of the number of integer solutions of
F(x,y)=0
except when the (projective completion of the) curve defined by
F(x,y)=0 has genus 0 and has at most 2 points at infinity. This
theorem generalizes also to Sintegers in algebraic number fields.
But, Siegel's theorem is ineffective. Bilu & Tichy obtained for the
equation f(x)=g(y) with f,g polynomials over ℚ, a
remarkable theorem which makes Siegel's theorem much more explicit
(although still ineffective). Here, we discuss some Diophantine
equations where BiluTichy theorem applies and some others where
elementary methods or methods of the Chebotarev density theorem
apply.
