In this talk, I will show that the multipliers at the cycles with periods 1 and 2 provide a good description of the space P_d of polynomial maps of degree d modulo conjugation by an affine transformation. More precisely, the elementary symmetric functions of the multipliers at the cycles with periods 1 and 2 induce a birational morphism from P_d onto its image. This result arises as the combination of the following two statements:
(1) Any sequence of complex polynomials of degree d with bounded multipliers at its cycles with periods 1 and 2 is necessarily bounded in P_d(C).
(2) A generic conjugacy class of complex polynomials of degree d is uniquely determined by its multipliers at its cycles with periods 1 and 2.
The second statement proves a conjecture by Hutz and Tepper and strengthens recent results by Ji and Xie in the particular case of polynomial maps.