The exceptional set for the projection from the moduli space of polynomials

The natural projection from the moduli space of polynomials of degree n is not surjective if n ≥ 4. We give explicit parametric representation of the exceptional set when n = 4 and 5. And we describe degeneration which occurs above the exceptional set when n = 4. Also we show that the preimage of a point generally consists of (n − 2)! points, where (n − 2)! is the maximum when the preimage is a finite set. 1 Known results Let Polyn be the space of all polynomial maps of degree n: p(z) = anz + an−1z + · · · + a1z + a0, (a j ∈ C ( j = 1, · · · , n), an , 0). Let A be the group of all affine transformations. We say that two maps p1, p2 ∈ Polyn are affine conjugate, denoted by p1 ∼A p2, if there exist a g ∈ A with g ◦ p1 ◦ g−1 = p2. The moduli space of polynomial maps degree n is the set of all affine conjugacy classes of elements in Polyn, which is denoted by Mn. For each f ∈ Polyn, let z1, z2, · · · , zn+1 be the fixed points of f and μ j the multipliers at z j; μ j = f ′(z j) (1 ≤ j ≤ n + 1), we set zn+1 = ∞ and hence μn+1 = 0. The elementary symmetric functions of μ j are σn,1 = μ1 + μ2 + · · · + μn+1, · · · , σn,r = ∑ j1< j2<···< jr μ j1μ j2 · · · μ jr , · · · , σn,n+1 = μ1μ2 · · · μn+1(= 0). (1) Note that these quantities are invariant under affine conjugacy.