Nonlinear Equations - UFRJ

Chapter 3
Topology and zero
counting
Arbitrarily small perturbations can obliterate zeros of
smooth, even analytic real functions. For instance, x 2 = 0 admits a
(double) root, but x 2 = ɛ admits no root for ɛ < 0.
This cannot happen for complex analytic mappings. Recall that
a real function ϕ from a metric space is lower semi-continuous at x
if and only if,
∀δ > 0, ∃ɛ > 0 s.t.(d(x, y) < ɛ) ⇒ ϕ(y) ≥ ϕ(x) − δ.
We will prove in Theorem 3.9) that the number of isolated roots
of an analytic mapping is lower semi-continuous. As the local root
count n U (f) = #{x ∈ U : f(x) = 0} is a discrete function, this just
means that
∃ɛ > 0 s.t. sup ‖f(x) − g(x)‖ < ɛ) ⇒ n U (y) ≥ n U (x).
x∈U
Gregorio Malajovich, Nonlinear equations.
28 o Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 2011.
Copyright c○ Gregorio Malajovich, 2011.
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