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Counter-Examples to Quantifier Elimination for Fewnomial and Exponential Expressions

By Andrei Gabrielov


Introduction. The famous Tarski-Seidenberg [10,11] theorem asserts that any real semialgebraic expression with quantifiers is equivalent to a real semialgebraic expression without quantifiers, i.e., quantifier elimination is possible for semialgebraic expressions. A well known example (Osgood [9]) shows that this result cannot be extended to expressions with the exponential function, even if all the variables remain bounded: an expression x, y, z, #u # [0, 1],y= xu, z = x exp(u) defines a germ Z of a subanalytic set at the origin in R 3 such that any analytic function (and even any formal power series) vanishing on Z should be identically zero. Gabrielov [3,4] showed the possibility of the bounded quantifier simplification for real semianalytic expressions i.e., equivalence of any such expression, with real an

Year: 1997
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