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Complex Analogues of the Rolle's Theorem

Sendov, Blagovest (2007)

Serdica Mathematical Journal

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.

Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

(2016)

Acta Arithmetica

For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≢ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p ) 1 / p , a j , such that ( x - 1 ) k divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let μ q ( n , L ) be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that | Q ( 0 ) | > 1 / L ( j = 1 n | Q ( j ) | q ) 1 / q . We find the size of κ p ( n , L ) and μ q ( n , L ) for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about μ ( n , L ) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special...

Criterion of the reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

Innocent Ndikubwayo (2020)

Czechoslovak Mathematical Journal

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence { P i } i = 1 generated by a three-term recurrence relation P i ( x ) + Q 1 ( x ) P i - 1 ( x ) + Q 2 ( x ) P i - 2 ( x ) = 0 with the standard initial conditions P 0 ( x ) = 1 , P - 1 ( x ) = 0 , where Q 1 ( x ) and Q 2 ( x ) are arbitrary real polynomials.

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