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

  • Authors:

  • M. Coste;M. F. Roy

  • Affiliations:

  • IRMAR, Université de Rennes I Rennes, France;IRMAR, Université de Rennes I Rennes, France

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 1988

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Abstract

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 @x 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 P^i of P at @x, 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).