Elsevier

Journal of Symbolic Computation

Volume 5, Issues 1–2, February–April 1988, Pages 121-129
Journal of Symbolic Computation

Thom's lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets

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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).

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