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##
A Real Nullstellensatz and Positivstellensatz
for the Semipolynomials over an Ordered Field.

A Real Nullstellensatz and Positivstellensatz
for the Semipolynomials over an Ordered Field.
pdf

with González-Vega L.

Journal of Pure and Applied Algebra ** 90** (1993), 167-188.
** Abstract :**
Let ** K **
be an ordered field and ** R ** its real closure. A
semipolynomial will be defined as a function from ** R **^{n} to
** R ** obtained by composition of polynomial functions and the
absolute value. Every semipolynomial can be defined as a
straight-line program containing only instructions with the
following type: "polynomial", "absolute value", "max" and "min" and
such a program will be called a semipolynomial expression. It will
be proved, using the ordinary Real Positivstellensatz, a general
Real Positivstellensatz concerning the semipolynomial expressions.
Using this semipolynomial version for the Real Positivstellensatz we
shall get as consequences a continuous and rational solution for
the 17-th Hilbert problem, rational and continuous
versions for several cases in the Real Positivstellensatz and
constructive proofs for several theorems concerning the algebra over
the real numbers.