Skip to main content Accessibility help
×
Home

ITERATION OF QUADRATIC POLYNOMIALS OVER FINITE FIELDS

  • D. R. Heath-Brown (a1)

Abstract

For a finite field of odd cardinality $q$ , we show that the sequence of iterates of $aX^{2}+c$ , starting at $0$ , always recurs after $O(q/\text{log}\log q)$ steps. For $X^{2}+1$ , the same is true for any starting value. We suggest that the traditional “birthday paradox” model is inappropriate for iterates of $X^{3}+c$ , when $q$ is 2 mod 3.

Copyright

References

Hide All
1. Browning, T. D. and Heath-Brown, D. R., Forms in many variables and differing degrees. J. Eur. Math. Soc. (JEMS) 19 2017, 357394.
2. Castelnuovo, G., Ricerche di geometria sulle curve algebriche. Atti Reale Accad. Sci. Torino 24 1889, 346373.
3. Juul, J., Kurlberg, P., Madhu, K. and Tucker, T. J., Wreath products and proportions of periodic points. Int. Math. Res. Not. IMRN 2016(13) 2016, 39443969.
4. Pollard, J. M., A Monte Carlo method for factorization. Nordisk Tidskr. Informationsbehandling (BIT) 15(3) 1975, 331334.
5. Shao, X., Polynomial values modulo primes on average and sharpness of the larger sieve. Algebra Number Theory 9(10) 2015, 23252346.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed