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Effective real nullstellensatz and variants

Effective Methods in Algebraic Geometry. Eds. Mora T., Traverso C.. Birkhaüser (1991).
Progress in Math. No94 (MEGA 90), 263-288.
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** Abstract :**
We give a constructive proof of the real
Nullstellensatz. So we obtain, for every ordered field **K**,
a uniformly primitive recursive algorithm that computes,
for the input ``a system of generalized signs
conditions ($gsc$) on polynomials of **K**[X_1,X_2,...,X_n]
impossible to satisfy in the real closure of
**K**, an algebraic identity that makes this impossibility
evident. The main idea is to give an
``algebraic identity version'' of universal and existential
axioms of the theory of real closed fields, and
of the simplest deduction rules of this theory (as Modus Ponens).
We apply this idea to the
Hörmander algorithm, which is the conceptually simplest test for
the impossibility of a $gsc$ system in the real closure of an ordered
field.