Finding zeroes of polynomials in the j-invariant and its derivatives

A family of conjectures inspired by Zilber's work predicts that polynomial equations in various special functions should have solutions, provided suitable geometric conditions are satisfied. I will talk of the very special case of one-variable equations in the j-invariant and its derivatives, where we prove that the zeroes are 'Zariski-dense' in an appropriate sense. For instance, we can give a very qualitative (and ineffective) description of the zeroes of j''. The proof builds on previous techniques used for exponentiation, rooted in elementary complex analysis, but also new tricks, such as zero estimates on SL₂(ℤ) orbits. This is joint work with Vahagn Aslanyan and Sebastian Eterović.