二次多項式的樹型伽洛瓦群表現理論

For a quadratic polynomial with integral coefficients, if all of iterates are irreducible over Q and it is not post-critically finite, enough results and examples suggest that the Galois group corresponding to the roots of iterates is sufficiently large. In fact, by assuming ABC conjecture, it can be shown that the index of the Galois group is of finite index in the automorphism group of the extended tree constructed by the roots of iterates. There is a system of quadratic polynomials which satisfy this property. In this project, we plan to find more quadratic polynomials which satisfy this property and moreover, we expect to prove this property for quadratic polynomials without using ABC conjecture. In this report, we find a new system of quadratic polynomials which satisfy this finite index property.