and its n-th derivative. Let the interval the interval in which we are looking for roots. We assume that , and . We construct the sequence from which we exclude the 0 element. Similarly we construct the sequence (a special treatment has to be applied for the zero element of , see [13]). Let the number of change of sign in and the number of change of sign in . Then the number of real roots of in , counted with their order of multiplicity, is or lower than this number by an even number.

Jean-Pierre Merlet 2012-12-20