International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 5, Pages 331-343
Local maxima of a random algebraic polynomial
Department of Mathematics, University of Ulster, Jordanstown, Co. Antrim, BT37 0QB, UK
Received 22 September 1999
Copyright © 2001 K. Farahmand and P. Hannigan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present a useful formula for the expected number of maxima of a
normal process that occur below a level . In the
derivation we assume chiefly that , and have, with probability one, continuous 1 dimensional
distributions and expected values of zero. The formula referred to
above is then used to find the expected number of maxima below the
level for the random algebraic polynomial. This result
highlights the very pronounced difference in the behaviour of the
random algebraic polynomial on the interval compared with
the intervals and . It is also shown
that the number of maxima below the zero level is no longer on the intervals and .