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.
Author(s): A. G. Khovanskii
Publisher: American Mathematical Society
Year: 1991
Language: English
Pages: 151
Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents ......Page 8
Introduction ......Page 10
CHAPTER I. An Analogue of the Bezout Theorem for a System of Real Elementary Equations ......Page 18
1. General estimate of the number of roots of a system of equations ......Page 19
2. Estimate of the number of solutions of a system of quasipolynomials ......Page 21
3. A version of the general estimate of the number of roots of a system of equations ......Page 22
4. Estimate of the number of solutions of a system of trigonometric quasipolynomials ......Page 23
5. Elementary functions of many real variables ......Page 25
6. Estimate of the number of solutions of a system of elementary equations ......Page 26
7. Remarks ......Page 27
1. Rolle's theorem for dynamical systems ......Page 30
2. Algebraic properties of P-curves ......Page 32
3. One more version of the theory of fewnomials ......Page 36
CHAPTER III. Analogues of the Theorems of Rolle and Bezout for Separating Solutions of Pfaff Equations ......Page 40
1. Coorientation and linking index ......Page 42
2. Separating submanifolds ......Page 45
3. Separating solutions of Pfaff equations ......Page 48
4. Separating solutions on 1-dimensional manifolds; an analogue of Rolle's theorem ......Page 51
5. Higher-dimensional analogues of Rolle's estimate ......Page 54
6. Ordered systems of Pfaff equations, their separating solutions and characteristic sequences ......Page 60
7. Estimate of the number of points in a zero-dimensional separating solution of an ordered system of Pfaff equations via the generalised number of zeroes of the characteristic sequence of the system ......Page 65
8. The virtual number of zeroes ......Page 71
9. Representative families of divisorial sequences ......Page 74
10. The virtual number of zeroes on a manifold equipped with a volume form ......Page 77
11. Estimate of the virtual number of zeroes of a sequence with isolated singular points via their order and index ......Page 82
12. A series of analogues of Rolle's estimate and Bezout's theorem ......Page 85
13. Fewnomials in a complex region and Newton polyhedra ......Page 90
14. Estimate of the number of connected components and the sum of the Betti numbers of higher-dimensional separating solutions ......Page 98
CHAPTER IV. Pfaff Manifolds ......Page 104
1. Simple afflne Pfaff manifolds ......Page 105
2. Affine Pfaff manifolds ......Page 107
3. Pfaff A-manifolds ......Page 112
4. Pfaff manifolds ......Page 116
5. Pfaff functions in Pfaff domains in R" ......Page 119
6. Results ......Page 121
CHAPTER V. Real-Analytic Varieties with Finiteness Properties and Complex Abelian Integrals ......Page 124
1. Basic analytic varieties ......Page 125
2. Analytic Pfaff manifolds ......Page 126
3. Finiteness theorems ......Page 128
4. Abelian integrals ......Page 129
Conclusion ......Page 132
Appendix. Pfaff equations and limit cycles, by Yu. S. II' yashenko ......Page 138
Bibliography ......Page 142
Index ......Page 146
