Number Theory Seminar: Himadri Ganguli

Abstract.

In recent past we were interested to study some special composition of polynomial equation f(g(x)) = f(x)hm(x) where f, g and h are unknown polynomials with coefficients in arbitrary field K, f is non-constant and separable, deg g ! 2, g! "= 0 and the integer power m ! 2 is not divisible
by the characteristic of the field K. In this talk we prove that this equation has no solutions if deg f ! 3. If deg f = 2, we prove that m = 2 and give all solutions explicitly in terms of Chebyshev polynomials. The diophantine applications for such polynomials f, g, h with coefficients in Q or Z are considered in the context of the conjecture of Cassaigne et. al. on the values of Louiville’s ! function at points f(r), r # Q. This is joint work with Jonas Jankauskas.