**Translations of Mathematical Monographs**

1991;
139 pp;
Hardcover

MSC: Primary 12;
Secondary 14; 58

**Print ISBN: 978-0-8218-4547-9
Product Code: MMONO/88**

List Price: $73.00

Individual Member Price: $58.40

# Fewnomials

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*A.G. Khovanskii*

The ideology of the theory of fewnomials is the following: real
varieties defined by “simple,” not cumbersome, systems of
equations should have a “simple” topology. One of the results of
the theory is a real transcendental analogue of the Bezout theorem: for a
large class of systems of \(k\) transcendental equations in
\(k\) real variables, the number of roots is finite and can be
explicitly estimated from above via the “complexity” of the
system. A more general result is the construction of a category of real
transcendental manifolds that resemble algebraic varieties in their properties.
These results give new information on level sets of elementary functions and
even on algebraic equations.

The topology of geometric objects given via algebraic equations
(real-algebraic curves, surfaces, singularities, etc.) quickly becomes more
complicated as the degree of the equations increases. It turns out that the
complexity of the topology depends not on the degree of the equations but
only on the number of monomials appearing in them. This book provides a
number of theorems estimating the complexity of the topology of geometric
objects via the cumbersomeness of the defining equations. In addition, the
author presents a version of the theory of fewnomials based on the model of a
dynamical system in the plane. Pfaff equations and Pfaff manifolds are also
studied.