International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 5, Pages 331-343

Local maxima of a random algebraic polynomial

K. Farahmand and P. Hannigan

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 ξ(t) that occur below a level u. In the derivation we assume chiefly that ξ(t),ξ(t), and ξ(t) 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 u for the random algebraic polynomial. This result highlights the very pronounced difference in the behaviour of the random algebraic polynomial on the interval (1,1) compared with the intervals (,1) and (1,). It is also shown that the number of maxima below the zero level is no longer O(logn) on the intervals (,1) and (1,).