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How to recognize zero? (English) Zbl 0917.11062
An elementary point is a point in complex $$n$$ space which is an isolated, nonsingular solution of $$n$$ equations in $$n$$ variables, each equation being either of the form $$p=0$$, where $$p$$ is a polynomial in $$\mathbb{Q}[x_1, \dots, x_n]$$, or of the form $$x_j-e^{x_i}=0$$. An elementary number is the polynomial image of an elementary point. The scope of this paper is to present a semi-algorithm to decide whether or not a given elementary number is zero. It is proved that this semi-algorithm is an algorithm, i.e. that it always terminates, unless it is given a problem containing a counterexample to Schanuel’s conjecture.

##### MSC:
 11U05 Decidability (number-theoretic aspects) 11J91 Transcendence theory of other special functions 11Y16 Number-theoretic algorithms; complexity 68W30 Symbolic computation and algebraic computation
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