Root separation for polynomials with reducible derivative
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Artūras Dubickas
Abstract
Suppose f is a degree d polynomial with integer coefficients whose derivative f′ is a polynomial reducible over ℚ. We give a lower bound for the distance between two distinct roots of f in terms of d, the height H(f) of f, and the degree m of the irreducible factor of f′ with largest degree. The exponent (d + m − 1)/2 that appears as the power of H(f) is smaller than the corresponding exponent d − 1 in Mahler’s bound.
(Communicated by István Gaál)
Acknowledgement
I thank both referees for noticing several misprints and suggesting some improvements. This research was funded by the European Social Fund according to the activity “Improvement of researchers’ qualification by implementing world-class R&D projects” of Measure No. 09.3.3-LMT-K-712-01-0037.
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© 2020 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Regular papers
- Monadic pseudo BE-algebras
- Quadruple construction of decomposable double MS-algebras
- Fibonacci numbers in generalized Pell sequences
- Solutions of a generalized markoff equation in Fibonacci numbers
- Root separation for polynomials with reducible derivative
- Quadratic refinements of Young type inequalities
- Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
- Integration with respect to deficient topological measures on locally compact spaces
- Differential subordinations and Pythagorean means
- Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables
- Unbounded oscillation of fourth order functional differential equations
- Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term
- Entropy as an integral operator: Erratum and modification
- Unital topology on a unital l-group
- On the topological complexity of Grassmann manifolds
- Properties and methods of estimation for a bivariate exponentiated Fréchet distribution
- Existence and Hyers-Ulam stability results for a class of fractional order delay differential equations with non-instantaneous impulses
- Two semigroup rings associated to a finite set of meromorphic functions
