Displaying similar documents to “A constructive proof of the Density of Algebraic Pfaff Equations without Algebraic Solutions”

Homology classes of real algebraic sets

Wojciech Kucharz (2008)

Annales de l’institut Fourier

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There is a large research program focused on comparison between algebraic and topological categories, whose origins go back to 1952 and the celebrated work of J. Nash on real algebraic manifolds. The present paper is a contribution to this program. It investigates the homology and cohomology classes represented by real algebraic sets. In particular, such classes are studied on algebraic models of smooth manifolds.

Semi-algebraic neighborhoods of closed semi-algebraic sets

Nicolas Dutertre (2009)

Annales de l’institut Fourier

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Given a closed (not necessarly compact) semi-algebraic set X in ℝ n , we construct a non-negative semi-algebraic 𝒞 2 function f such that X = f - 1 ( 0 ) and such that for δ > 0 sufficiently small, the inclusion of X in f - 1 ( [ 0 , δ ] ) is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of  X .

An explicit formula for period determinant

Alexey A. Glutsyuk (2006)

Annales de l’institut Fourier

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We consider a generic complex polynomial in two variables and a basis in the first homology group of a nonsingular level curve. We take an arbitrary tuple of homogeneous polynomial 1-forms of appropriate degrees so that their integrals over the basic cycles form a square matrix (of multivalued analytic functions of the level value). We give an explicit formula for the determinant of this matrix.

On algebraic solutions of algebraic Pfaff equations

Henryk Żołądek (1995)

Studia Mathematica

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We give a new proof of Jouanolou’s theorem about non-existence of algebraic solutions to the system ẋ = z s , ẏ = x s , ż = y s . We also present some generalizations of the results of Darboux and Jouanolou about algebraic Pfaff forms with algebraic solutions.

Finite automata and algebraic extensions of function fields

Kiran S. Kedlaya (2006)

Journal de Théorie des Nombres de Bordeaux

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We give an automata-theoretic description of the algebraic closure of the rational function field 𝔽 q ( t ) over a finite field 𝔽 q , generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over 𝔽 q . In passing, we obtain a characterization of well-ordered sets of rational numbers whose base p expansions are generated by a finite automaton, and exhibit some techniques for computing in the algebraic closure; these include an adaptation...