On the equation of degree 6
Let \({\mathcal P}_m\) denote the set of monic univariate polynomials of degree \(m\) with complex coefficients and distinct roots. Define \(\Delta:=\{(z_1,z_2,\ldots,z_m)\in{\mathbb C}^m\mid z_i\not=z_j\, \forall i\not=j\}\). The symmetric group, \(S_m\), acts on \({\mathbb C}^m\setminus\Delta\) by permuting entries. \({\mathcal P}_m\) can be identified with the set of orbits \(\left({\mathbb C}^m\setminus\Delta\right)/S_m\). The resulting covering, \(\gamma_m: {\mathbb C}^m\setminus\Delta \to {\mathcal P}_m\), constructs a polynomial from its roots. To factor a polynomial \(f\in{\mathcal P}_m\), it is sufficient to find a local cross section of \(\gamma_m\) defined on a neighbourhood of \(f\). The \textit{Schwarz genus} of \(\gamma_m\) is, by definition, the smallest integer \(g\) such that there exists an open cover \(\{U_1,\ldots,U_g\}\) of \({\mathcal P}_m\) such that for each \(i\) the restriction of \(\gamma_m\) to \(U_i\) is trivial. Thus the Schwarz genus gives the minimum number of local cross sections required to give a factorisation algorithm. By a general dimension argument, the Schwarz genus of \(\gamma_m\) is less than or equal to \(m\). From the work of \textit{V. A.~Vassiliev} [Complements of discriminants of smooth maps: topology and applications (Translated from the Russian by B.~Goldfarb, Translations of Mathematical Monographs 98, American Mathematical Society, Providence, RI) (1992; Zbl 0762.55001)], equality holds if \(m\) is a power of a prime. Thus the first interesting case is \(m=6\). The primary result of this paper is that the genus of \(\gamma_6\) is \(5\). The proof relies on obstruction theory and reduces to a group homology calculation.
scientific article; zbMATH DE number 2113907