A quartic Thue equation is a Diophantine equation F(x, y) = m in unkown integers x and y, where F is a given binary quartic form with integer coefficients and m is a given non-zero integer.

In this talk, I will introduce a research on upper bound on the number of solutions to quartic Thue equations.

We encounter with classification problem of binary quartic forms with integer coefficients modulo unimodular change of variables. This can be solved by combining reduction theory found in Cremona's paper and Baker theory.