On Topological Invariants of Real Algebraic Functions

We consider a natural covering responsible for the complexity of the ramification of roots of the general real polynomial equation, and calculate the homology groups of its base; for equations of degree ≤ 5 we give a complete description of the topology of this base. The general complex d-valued entire algebraic function x = F (a1, . . . , ad), given by the equation (1) x + a1x + · · ·+ ad−1x+ ad = 0, is ramified at the discriminant variety ΣC ⊂ C, consisting of collections of coefficients (a1, . . . , ad), for which the polynomial (1) has multiple roots, see. [1], [5]. The complement of this variety in C is the base of two standard coverings: dfold and d!-fold ones. The fundamental group of this complement acts on the sets of roots of the equation (1) and generates the entire permutation group of these roots. V.I. Arnold has exploited the homology classes of this complement C \ ΣC as obstructions to inducing one algebraic functions from the others. Also, the study of these homology groups provides lower estimates on the Schwarz genus of corresponding coverings, i.e. on the minimal number of open subsets covering the base, over any of which the covering has a continuous section, see [7], [8]. In [8], [9] these estimates are applied to the study of the topological complexity of approximate solution of the general equation (1). If we consider only real equations (1), then a similar role will be played by coverings defined on the complement of a certain subset of real codimension 2 in the space R of such equations. Namely, this subset Υ consists of polynomials having either a real root of multiplicity ≥ 3, or a couple of imaginary complex conjugate roots of multiplicity ≥ 2. For d = 4 this set in the space of reduced (i.e. with a1 = 0) polynomials (1) is represented by three branches of curves, going from the origin to the infinity and distinguished in the left-hand part of Fig. 1: two branches of the cuspidal edge of the swallowtail (see e.g. [2]) and the continuation of its self-intersection line. The monodromy of these coverings generates not the entire permutation group of d roots, but only the subgroup of even permutations. As in the complex case, the topology of these coverings provides lower estimates on the numbers of branchings of algorithms solving real equations (1); already for d = 3 this calculation proves the necessity of such branchings. In §4 we calculate integral homology groups of all spaces R \Υ. Supported by grant NSh-8462.2010.1 of President of Russia for the support of leading scientific schools. 1