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Thom’s lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets. (English) Zbl 0689.14006
Summary: Thom’s lemma, a very simple and basic result in real algebraic geometry, and explained in section 1, has a lot of interesting computational consequences. We shall outline two of these.
The first one is the fact that a real root \(\xi\) 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 \(\xi\), \(i=1,...,n-1\). 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 semialgebraic set where the incidence relation between the cells is easily obtained (see section 3).

MSC:
14Pxx Real algebraic and real-analytic geometry
94B40 Arithmetic codes
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