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Hurwitz reduced this to a combinatorial problem and conjectured the solution for rational functions with one degenerate critical level (proven combinatorially in 1996 by Goulden and Jackson [?] and Strehl [?], and from the singularity point of view in 1997 by Goryunov and Lando [?]). Lando and Zvonkine [?] solved the problem for non-Morse polynomials with fixed degenerate critical levels and we obtain a solution for even and odd polynomials. Our main tool, as it is in [?], is the Lyashko-Looijenga, LL, mapping which associates to a polynomial of degree n the unordered set of its n − 1 critical values. In particular we calculate the number of topological types of even and odd polynomials in a given discriminant stratum. The discriminant of a family of polynomials is the set of polynomials with multiple roots. Within this set lie various strata determined by the distribution of critical points on fixed critical levels. We call these strata discriminant strata and we define them using passports . The number of topological types that we are seeking is closely related to the degree of the LL mapping on the discriminant strata.

The multiplicity of the Lyashko-Looijenga mapping on the discriminant strata of even and odd polynomials