Case distinctions are necessary for representing polynomials as sums of squares

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This chapter provides an overview of the history of logical aspects of Hilbert's 17th problem. In his book on the foundations of geometry, Hilbert described those problems in plane geometrical construction that can be solved by means of only his five groups of axioms, can always be carried out by the use of straightedge and gauge. He gave two algebraic characterizations of the set of points so constructible, in terms of their Cartesian coordinates (f1 (x), f2(x)), where the given points are expressed as rational functions of the parameters x = (x0, …, xn) ε ℜn+1. The second of his two characterizations was a necessary and sufficient condition—namely that fi (x) be a totally real-algebraic number for all x ε Pn+1. The chapter discusses two logical points concerning the finiteness theorem: (1) intuitionistic considerations and (2) the terminology finiteness. © 1982, North-Holland Publishing Company

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Studies in Logic and the Foundations of Mathematics

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