eBook ISBN:  9781470445003 
Product Code:  MMONO/88.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
eBook ISBN:  9781470445003 
Product Code:  MMONO/88.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 

Book DetailsTranslations of Mathematical MonographsVolume: 88; 1991; 139 ppMSC: Primary 12; Secondary 14; 58
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 (realalgebraic 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

Chapters

Introduction

Chapter I. An Analogue of the Bezout Theorem for a System of Real Elementary Equations

Chapter II. Two Simple Versions of the Theory of Fewnomials

Chapter III. Analogues of the Theorems of Rolle and Bezout for Separating Solutions of Pfaff Equations

Chapter IV. Pfaff Manifolds

Chapter V. RealAnalytic Varieties with Finiteness Properties and Complex Abelian Integrals

Conclusion


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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 (realalgebraic 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.

Chapters

Introduction

Chapter I. An Analogue of the Bezout Theorem for a System of Real Elementary Equations

Chapter II. Two Simple Versions of the Theory of Fewnomials

Chapter III. Analogues of the Theorems of Rolle and Bezout for Separating Solutions of Pfaff Equations

Chapter IV. Pfaff Manifolds

Chapter V. RealAnalytic Varieties with Finiteness Properties and Complex Abelian Integrals

Conclusion