Displaying similar documents to “Traces of monomials in algebraic numbers”

Smale's Conjecture on Mean Values of Polynomials and Electrostatics

Dimitrov, Dimitar (2007)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35. A challenging conjecture of Stephen Smale on geometry of polynomials is under discussion. We consider an interpretation which turns out to be an interesting problem on equilibrium of an electrostatic field that obeys the law of the logarithmic potential. This interplay allows us to study the quantities that appear in Smale’s conjecture for polynomials whose zeros belong to certain specific regions. A conjecture...

Generalization of a Conjecture in the Geometry of Polynomials

Sendov, Bl. (2002)

Serdica Mathematical Journal

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In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by...

A deformation of commutative polynomial algebras in even numbers of variables

Wenhua Zhao (2010)

Open Mathematics

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We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian...

Complex Analogues of the Rolle's Theorem

Sendov, Blagovest (2007)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 30C10. Classical Rolle’s theorem and its analogues for complex algebraic polynomials are discussed. A complex Rolle’s theorem is conjectured.