On the Number of Rational Iterated Pre-Images of -1 under Quadratic Dynamical Systems

Hyde, Trevor

doi:10.33697/ajur.2010.010

Cited by 2 publications

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“…The cases of a = 0 and a = −1 were studied in [5,9], respectively, and it was shown that κ(0, Q) = κ(−1, Q) = 6. In [5] a significant amount of effort went into the more difficult task of showing that κ(0, Q) = 6, assuming some standard conjectures.…”

Section: Introductionmentioning

confidence: 99%

Preimages of quadratic dynamical systems

Hutz

Hyde

Krause

2011

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(Communicated by Bjorn Poonen)For a quadratic polynomial with rational coefficients, we consider the problem of bounding the number of rational points that eventually land at a given constant after iteration, called preimages of the constant. It was shown by Faber, Hutz, Ingram, Jones, Manes, Tucker, and Zieve (2009) that the number of rational preimages is bounded as one varies the polynomial. Explicit bounds on the number of preimages of zero and −1 were addressed in subsequent articles. This article addresses explicit bounds on the number of preimages of any algebraic number for quadratic dynamical systems and provides insight into the geometric surfaces parameterizing such preimages.

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Section: Introductionmentioning

confidence: 99%

Preimages of quadratic dynamical systems

Hutz

Hyde

Krause

2011

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“…The cases of a = 0 and a = −1 were studied in [Faber et al 2011;Hyde 2010], respectively, and it was shown that…”

Section: Introductionmentioning

confidence: 99%

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2011

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For fixed a and b, let Q n be the family of polynomials q(x) all of whose roots are real numbers in [a, b] (possibly repeated), and such that q(a) = q(b) = 0. Since an element of Q n is completely determined by it roots (with multiplicity), we may ask how the polynomial is sensitive to changes in the location of its roots. It has been shown that one of the Bernstein polynomials b i (x) = (x −a) n−i (x −b) i , i = 1,. .. , n − 1, is the member of Q n with largest supremum norm in [a, b]. Here we show that for p ≥ 1, b 1 (x) and b n−1 (x) are the members of Q n that maximize the L p norm in [a, b]. We then find the associated maximum values. MSC2000: 30C15.

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On the Number of Rational Iterated Pre-Images of -1 under Quadratic Dynamical Systems

Abstract: For the class of functions fc(x)=x^2+c, we prove a conditional bound on the number of rational solutions to fc^N(x) =−1 and make computational conjectures for a bound on the number of rational solutions to fc^N(x)=a fora in a specific subset of the rationals.

Cited by 2 publications

References 7 publications

Preimages of quadratic dynamical systems

Preimages of quadratic dynamical systems

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