Thom's lemma, a very simple and basic result in real algebraic geometry, and explained in section I, has a lot of interesting computational consequences.
We shall outline two of these.
The first one is the fact that a real root ξ of a polynomial P of degree n with real coefficients may be distinguished from the other real roots of P by the signs of the derivatives Pi of P at ξ, i = 1, ..., n - 1. This offers a new possibility for the coding of real algebraic numbers and for computation with these numbers (see section 2).
The second is based on a generalisation of Thom's lemma to the case of several variables. It gives, after a linear change of coordinates, a cylindric algebraic decomposition of a semi- algebraic set where the incidence relation between the cells is easily obtained (see section 3).