Resolutions of discriminants and topology of their complements

Abstract

Abstract. We study topological invariants of spaces of nonsingular geometrical objects (such as knots, operators, functions, varieties) defined by the linking numbers with appropriate cycles in the complementary discriminant sets of degenerate objects. We describe the main construction of such classes (based on the conical resolutions of discriminants) and list the results for a number of examples.The discriminant subsets of spaces of geometric objects are the sets of all objects with singularities of some chosen type. The important examples are: spaces of polynomials with multiple roots, resultant sets of polynomial systems having common roots, spaces of functions with degenerate singular points, of non-smooth algebraic varieties, of linear operators with zero or multiple eigenvalues, of smooth maps S 1 → M n (n ≥ 3) having singular or self-intersection points, of non-generic plane curves, and many others.The discriminants are usually singular varieties, whose stratifications correspond to the classification of degenerations of the corresponding objects. E.g., the discriminant subset in the space of polynomials x 3 + ax + b is the semicubical parabola (a/3) 3 + (b/2) 2 = 0: its regular points correspond to polynomials with a root of multiplicity exactly 2, and the vertex to the polynomial x 3 . The discriminant in the space of polynomials x 4 +ax 2 +bx+c is the swallowtail, i.e. the surface shown in the right-hand part of Fig. 1: its self-intersection curve consists of polynomials having two double roots, and the semicubical edges correspond to the polynomials with one triple root; the most singular point is the polynomial x 4 with a root of multiplicity 4. Similar stratifications hold for polynomials of all higher degrees: their strata are indexed by the multiplicities and orders in R 1 of all corresponding multiple roots.Usually one is interested in the space of non-singular objects which is the complement of the discriminant Σ, e.g. in the space of polynomials without multiple roots, of smooth varieties, of non-degenerate operators, or of knots, i.e. maps S 1 → R 3 having no self-intersection or singular points.If the total space F of geometric objects is an N -dimensional vector space then the homology groups of these complementary spaces are related by the Alexander duality formula

Special 5-term recurrence relations, Banded Toeplitz matrices, and Reality of Zeros

“…It is a surface which self-intersects along a curve corresponding to polynomials having two double real zeros, the cuspidal cubic edges correspond to the polynomials with one triple zero while the most singular point is the quartic polynomial x 4 with a zero of multiplicity 4. For the spaces of polynomials of higher degrees, a similar strata will also appear [11]. For a given c = (c 1 , c 2 , c 3 ) ∈ R 3 , one can determine in which region of the complement of D(F) the point c lies, by checking if the set of real zeros of the polynomial has 0, 2 or 4 elements:-these being the only possibilities for a real quartic polynomial, as non-real zeros appear in pairs as complex conjugates.…”

Homology of non-$k$-overlapping discs

“…But at the same time they differ from Imm (k) (M, R n ) and Imm There are two main approaches to study such functional spaces. The first approach, due to Vassiliev and usually called Theory of Discriminants [43], consists in considering the space of all smooth maps from our manifold to R n . This space is an affine space of infinite dimension and thus contractible.…”

How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties

“…Now we would like to make a few concluding remarks about topology of the complex strata {D µ } and {D • µ }. A wonderfully rich account of the topological properties of disciminants, or rather their complements, can be found in [Va1], [Va2]. Both sources concentrate on more subtle description of real deteminantal varieties.…”

“…properties of disciminants, or rather their complements, can be found in [Va1], [Va2]. Both sources concentrate on more subtle description of real deteminantal varieties.…”