The statistics of continued fractions for polynomials over a finite field |
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| Authors: | Christian Friesen Doug Hensley |
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| Affiliation: | Department of Mathematics, Ohio State University, Marion Campus, Marion, Ohio 43302 ; Department of Mathematics, Texas A& M University, College Station, Texas 77843 |
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| Abstract: | Given a finite field  of order  and polynomials ![$a,bin F[X]$](http://www.ams.org/proc/1996-124-09/S0002-9939-96-03394-1/gif-abstract/img5.gif) of degrees  respectively, there is the continued fraction representation  . Let  denote the number of such pairs for which  and for   . We give both an exact recurrence relation, and an asymptotic analysis, for  . The polynomial associated with the recurrence relation turns out to be of P-V type. We also study the distribution of  . Averaged over all  and  as above, this presents no difficulties. The average value of  is  , and there is full information about the distribution. When  is fixed and only  is allowed to vary, we show that this is still the average. Moreover, few pairs give a value of  that differs from this average by more than  |
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