Nonlinear Equations - UFRJ

Chapter 1
Counting solutions of
polynomial systems
In this notes, we will mostly look at equations over the field
of complex numbers. The case of real equations is interesting but
more difficult to handle. In many situations, it may be convenient to
count or to solve over C rather than over R, and then ignore non-real
solutions.
Finding or even counting the solutions of specific systems of polynomials
is hard in the complexity theory sense. Therefore, instead
of looking at particular equations, we consider linear spaces of equations.
Several bounds for the number of roots are known to be true
generically. As many definitions of genericity are in use, we should
be more specific.
Definition 1.1 (Zariski topology). A set V ⊆ C N is Zariski closed
Gregorio Malajovich, Nonlinear equations.
28 o Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 2011.
Copyright c○ Gregorio Malajovich, 2011.
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