We investigate necessary and sufficient conditions for an arbitrary polynomial of degree n to be trivial, i.e. to have the form a(z − b) n. These results are related to an open problem, conjectured in 2001 by E. Casas-Alvero. It says, that any complex univariate polynomial, having a common root with each of its non-constant derivative must be a power of a linear polynomial. In particular, we establish determinantal representation of the Abel-Goncharov interpolation polynomials, related to the problem and having its own interest. Among other results are new Sz.-Nagy type identities for complex roots and a generalization of the Schoenberg conjectured analog of Rolle's theorem for polynomials with real and complex coefficients.

Abstract. We show the existence of surfaces of degree d in È 3 ( ) with approximately 3j+2 6j(j+1) d 3 singularities of type A j , 2 ≤ j ≤ d − 1. The result is based on Chmutov's construction of nodal surfaces. For the proof we use plane trees related to the theory of Dessins d'Enfants.Our examples improve the previously known lower bounds for the maximum number µ A j (d) of A j -singularities on a surface of degree d in most cases. We also give a generalization to higher dimensions which leads to new lower bounds even in the case of nodal hypersurfaces in È n , n ≥ 5.To conclude, we work out in detail a classical idea of B. Segre which leads to some interesting examples, e.g. to a sextic with 36 cusps. IntroductionAll possible configurations of singularities on a surface of degree 3 in È 3 := È 3 ( ) are known since Schläfli's work [25] in the 19 th century, see [14] and [18] for explicit equations and illustrating pictures. In the case of degree 4, the classification was completed recently by Yang [29] using computers.Much less is known for higher degrees, even when restricting to a particular type of singularity. E.g., the maximum number of A 1 -singularities on a surface of degree d is only known for d ≤ 6. We recently improved the case d = 7 using computer algebra and geometry over prime fields [19]. The best lower bounds for surfaces of large degree d with A 1 -singularities are given by Chmutov's construction [9].For higher singularities -e.g., singularities of type A j which are locally equivalent to x j+1 + y 2 + z 2 -the situation is even more difficult. We denote by µ Aj (d) the maximum number of singularities of type A j a surface of degree d in È 3 can have. Barth [5] constructed a quintic with 15 singularities of type A 2 (also called (ordinary) cusps), and the author constructed a sextic with 35 such singularities [20] using computer algebra in characteristic zero which showed µ A2 (6) ≥ 35. The detailed study of a generalization of an idea of B. Segre [26] which we give in the appendix leads to: µ A2 (6) ≥ 36 (see equations (11) and (12), and corollary 12). Recently, Barth and others considered the codes connected to surfaces with threedivisible cusps in analogy to the codes related to even sets of nodes, see [4,6,23].In general, the best lower bounds for the µ Aj

Using explicit constructions of certain real line arrangements we show that Chmutov's construction can be adapted to give only real singularities. All currently best known constructions which exceed Chmutov's lower bound (i.e., for d = 3, 4, · · · , 8, 10, 12) can also be realized with only real singularities. Thus, our result shows that, up to now, all known lower bounds for µ(d) can be attained with only real singularities.We conclude with an application of the theory of real line arrangements which shows that our arrangements are aymptotically the best possible ones. This proves a special case of a conjecture of Chmutov.

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