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A real nullstellensatz and positivstellensatz for the semipolynomials over an ordered field.

Laureano González-Vega, Henri Lombardi (1992)

Extracta Mathematicae

Let K be an ordered field and R its real closure. A semipolynomial will be defined as a function from Rn 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, sup and inf and such a program will be called a semipolynomial expression. It will be proved, using the ordinary real positivstellensatz, a general real positivstellensatz concerning...

Embedding theorems for spaces of ℝ-places of rational function fields and their products

Katarzyna Kuhlmann, Franz-Viktor Kuhlmann (2012)

Fundamenta Mathematicae

We study spaces M(R(y)) of ℝ-places of rational function fields R(y) in one variable. For extensions F|R of formally real fields, with R real closed and satisfying a natural condition, we find embeddings of M(R(y)) in M(F(y)) and prove uniqueness results. Further, we study embeddings of products of spaces of the form M(F(y)) in spaces of ℝ-places of rational function fields in several variables. Our results uncover rather unexpected obstacles to a positive solution of the open question whether the...

Fields of surreal numbers and exponentiation

Lou van den Dries, Philip Ehrlich (2001)

Fundamenta Mathematicae

We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case,...

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