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**Fewnomials. Transl. from the Russian by Smilka Zdravkovska.**
*(English)*
Zbl 0728.12002

Translations of Mathematical Monographs, 88. Providence, RI: American Mathematical Society (AMS). viii, 139 p. $ 96.00 (1991).

In this monograph various estimates for the number of solutions of systems of real valued functions are obtained. The term “fewnomials” refers to polynomial functions with a small number of monomials, but in the book other classes of functions are also considered. The topic originates in a problem of A.G. Kushnirenko from the seventies and it was developed by the author in his doctoral dissertation.

The author’s genuine methods appeal to algebraic, geometric, topological and analytic techniques. In the sequel we mention some of the finiteness results from the book. It is proved that the number of nondegenerate roots of a polynomial system from a positive orthant is bounded by a function which depends on the number of significant monomials and on the number of equations. A similar bound is obtained for the number of connected components of a real algebraic set and estimates in function of Newton polyhedra are given. There are described the corresponding results for polynomials in the coordinates in \({\mathbb{R}}^ n\) and in the functions exp \(a_ i(x)\), \(i=1,2,...,q\), where \(a_ i\) are linear forms on \({\mathbb{R}}^ n\). We also mention an analogue of the theorem of Bézout for Pfaffian curves, some finiteness theorems for Pfaffian manifolds, the study of the real-analytic varieties with finiteness properties and a result about abelian integrals. Connections with other topics such as additive complexity of the polynomials, Tarski’s problem, Hilbert’s 16th problem, Hardy fields are described. The book contains an appendix about Pfaffian equations and limit cycles written by Yu. S. Il’yashenko.

The author’s genuine methods appeal to algebraic, geometric, topological and analytic techniques. In the sequel we mention some of the finiteness results from the book. It is proved that the number of nondegenerate roots of a polynomial system from a positive orthant is bounded by a function which depends on the number of significant monomials and on the number of equations. A similar bound is obtained for the number of connected components of a real algebraic set and estimates in function of Newton polyhedra are given. There are described the corresponding results for polynomials in the coordinates in \({\mathbb{R}}^ n\) and in the functions exp \(a_ i(x)\), \(i=1,2,...,q\), where \(a_ i\) are linear forms on \({\mathbb{R}}^ n\). We also mention an analogue of the theorem of Bézout for Pfaffian curves, some finiteness theorems for Pfaffian manifolds, the study of the real-analytic varieties with finiteness properties and a result about abelian integrals. Connections with other topics such as additive complexity of the polynomials, Tarski’s problem, Hilbert’s 16th problem, Hardy fields are described. The book contains an appendix about Pfaffian equations and limit cycles written by Yu. S. Il’yashenko.

Reviewer: D.Ştefănescu (Bucureşti)

### MSC:

12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

32B20 | Semi-analytic sets, subanalytic sets, and generalizations |

14P05 | Real algebraic sets |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14P15 | Real-analytic and semi-analytic sets |

58A17 | Pfaffian systems |

14A10 | Varieties and morphisms |

58C05 | Real-valued functions on manifolds |