On the real roots of Euler polynomials

We show how are located the positive roots of the Euler polynomiale _n of degreen. We give an upper bound and a lower bound for the greatest root. This permits to determine an integerv _(n) such that the number of positive roots ofE _n is eitherv _(n) orv _(n) +2. We also study the behaviour of ther-th positive root ofE _n asn tends to infinity.