On equivalence of ideals of real global analytic functions and the 17th Hilbert problem

1981 ◽  
Vol 63 (3) ◽  
pp. 403-421 ◽  
Author(s):  
Jacek Bochnak ◽  
Wojciech Kucharz ◽  
Masahiro Shiota
Keyword(s):  
Analytic Functions ◽  
Hilbert Problem ◽  
17Th Hilbert Problem

ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

2012 ◽  
Vol 23 (04) ◽  
pp. 1250031 ◽  
Author(s):  
JOSÉ F. FERNANDO ◽  
J. M. GAMBOA
Keyword(s):  
Integral Domain ◽  
Hilbert Problem ◽  
Semialgebraic Sets ◽  
Semialgebraic Set ◽  
Irreducible Components ◽  
Real Geometry ◽  
Nash Manifold ◽  
Substitution Theorem ◽  
Satisfactory Theory ◽  
17Th Hilbert Problem

In this work we define a semialgebraic set S ⊂ ℝn to be irreducible if the noetherian ring [Formula: see text] of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring [Formula: see text]: Substitution Theorem, Positivstellensätze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

scholarly journals On the Hilbert problem for analytic functions in quasihyperbolic domains

2019 ◽  
Vol 2 ◽  
pp. 23-30 ◽  
Author(s):  
V.Ya. Gutlyanskii ◽  
◽  
V.I. Ryazanov ◽  
E. Yakubov ◽  
A.S. Yefimushkin ◽  
...  
Keyword(s):  
Analytic Functions ◽  
Hilbert Problem

Real algebraic geometry and the 17th Hilbert problem

1980 ◽  
Vol 251 (3) ◽  
pp. 213-241 ◽  
Author(s):  
Jacek Bochnak ◽  
Gustave Efroymson
Keyword(s):  
Algebraic Geometry ◽  
Hilbert Problem ◽  
Real Algebraic Geometry ◽  
17Th Hilbert Problem

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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