Expansions of quadratic maps in prime fields
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- Proc. Amer. Math. Soc. 142 (2014), 85-92 Request permission
Abstract:
Let $f(x)=ax^2+bx+c\in \mathbb Z[x]$ be a quadratic polynomial with $a\not \equiv 0$ mod $p$. Take $z\in \mathbb F_p$ and let $\mathcal O_z=\{f_i(z)\}_{i\in \mathbb Z^+}$ be the orbit of $z$ under $f$, where $f_i(z)=f(f_{i-1}(z))$ and $f_0(z)=z$. For $M< |\mathcal O_z|$, we study the diameter of the partial orbit $\mathcal O_M=\{z, f(z), f_2(z),\dots , f_{M-1}(z)\}$ and prove that there exists $c_1>0$ such that \[ \operatorname {diam} \mathcal O_M \gtrsim \min \bigg \{ Mp^{\;{c_1}}, \frac 1{\log p} M^{\frac 45} p^{\frac 15}, M^{\;\frac 1{13}\log \log M}\bigg \}. \] For a complete orbit $\mathcal C$, we prove that \[ \operatorname {diam} \mathcal C \gtrsim \min \{p^{\;5c_1}, e^{ \;T/4}\;\},\] where $T$ is the period of the orbit.References
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Additional Information
- Mei-Chu Chang
- Affiliation: Department of Mathematics, University of California, Riverside, California 92521
- Email: mcc@math.ucr.edu
- Received by editor(s): October 10, 2011
- Received by editor(s) in revised form: March 2, 2012
- Published electronically: September 26, 2013
- Additional Notes: The author’s research was partially financed by the National Science Foundation.
- Communicated by: Bryna Kra
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 85-92
- MSC (2010): Primary 11B50, 37A45, 11B75; Secondary 11T23, 37F10, 11G99
- DOI: https://doi.org/10.1090/S0002-9939-2013-11740-5
- MathSciNet review: 3119183