# 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.

#### Table of Contents

# Table of Contents

## Fewnomials

- Cover Cover11
- Title page iii4
- Dedication v6
- Contents vii8
- Introduction 110
- Chapter I. An Analogue of the Bezout Theorem for a System of Real Elementary Equations 918
- Chapter II. Two Simple Versions of the Theory of Fewnomials 2130
- Chapter III. Analogues of the Theorems of Rolle and Bezout for Separating Solutions of Pfaff Equations 3140
- Chapter IV. Pfaff Manifolds 95104
- Chapter V. Real-Analytic Varieties with Finiteness Properties and Complex Abelian Integrals 115124
- Conclusion 123132
- Appendix 129138
- Bibliography 133142
- Subject index 137146
- Back Cover Back Cover1151