Displaying similar documents to “Vieta’s Formula about the Sum of Roots of Polynomials”

Special isomorphisms of F [ x 1 , ... , x n ] preserving GCD and their use

Ladislav Skula (2009)

Czechoslovak Mathematical Journal

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On the ring R = F [ x 1 , ⋯ , x n ] of polynomials in n variables over a field F special isomorphisms A ’s of R into R are defined which preserve the greatest common divisor of two polynomials. The ring R is extended to the ring S = F [ [ x 1 , ⋯ , x n ] ] + and the ring T = F [ [ x 1 , ⋯ , x n ] ] of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms A ’s are extended to automorphisms B ’s of the ring S . Using the property that the isomorphisms A ’s preserve...

On realizability of sign patterns by real polynomials

Vladimir Kostov (2018)

Czechoslovak Mathematical Journal

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The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers ( p , n ) , chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree 8 polynomials. ...

A note on k-c-semistratifiable spaces and strong β -spaces

Li-Xia Wang, Liang-Xue Peng (2011)

Mathematica Bohemica

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Recall that a space X is a c-semistratifiable (CSS) space, if the compact sets of X are G δ -sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a T 2 -space X is a k-c-semistratifiable space if and only if X has a g function which satisfies the following conditions: (1) For each x ∈ X , { x } = ⋂ { g ( x , n ) : n ∈ ℕ } and g ( x , n + 1 ) ⊆ g ( x , n ) for each...

Symmetric identity for polynomial sequences satisfying A n + 1 ' ( x ) = ( n + 1 ) A n ( x )

Farid Bencherif, Rachid Boumahdi, Tarek Garici (2021)

Communications in Mathematics

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Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A n + 1 ' ( x ) = ( n + 1 ) A n ( x ) with A 0 ( x ) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, ApostolEuler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

Global behavior of a third order rational difference equation

Raafat Abo-Zeid (2014)

Mathematica Bohemica

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In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation x n + 1 = a x n x n - 1 - b x n + c x n - 2 , n ∈ ℕ 0 where a , b , c are positive real numbers and the initial conditions x - 2 , x - 1 , x 0 are real numbers. We show that every admissible solution of that equation converges to zero if either a < c or a > c with ( a - c ) / b < 1 . When a > c with ( a - c ) / b > 1 , we prove that every admissible solution is unbounded. Finally, when a = c , we prove that every admissible solution converges to zero.