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Let A be the ring of polynomials in n indeterminates over ℝ. Then any subset S of ℝ n which is the solution set of a polynomial system ƒ1(x) > 0, …, ƒ k (x) > 0 is also the solution set of a system of n inequalities g1(x) > 0, … g n (x) > 0, no matter how big k is. This observation, made about twelve years ago for n ≤ 3 and proved in full generality five years later is the starting point of the present book.
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