Local maxima of a random algebraic polynomial

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 ( log n ) on the intervals ( − ∞ , − 1 ) and ( 1 , ∞ ) .