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Reduction of unknowns in diophantine representations. (English) Zbl 0773.11077
Authors’ summary: The hardest step to solve Hilbert’s tenth problem is to prove that the exponential relation is diophantine. In the study of decision problems concerning the solvability of diophantine equations with few unknowns, reducing unknowns in diophantine representations plays an important role. In this paper, we give diophantine representations of $C=\psi\sb B(A,1)$ (where $\psi\sb 0(A,1)=0$, $\psi\sb 1(A,1)=1$, $\psi\sb{m+1}(A,1)=A\psi\sb m(A,1)-\psi\sb{m-1}(A,1))$ and $W=V\sp B \wedge A\sb 1,\dots,A\sb k\in\square\wedge$ $S \vert T \wedge R > 0$ with only 3 and 5 natural number unknowns respectively, $C= \psi\sb B(A,1)$ (on the condition $1< \vert B \vert <{\vert A \vert \over 2}-1)$ and $W=V\sp B \wedge A\sb 1, \dots, A\sb k \in \square \wedge S \vert T$ with 4 and 6 integer unknowns respectively.

11U05Decidability related to number theory
03B25Decidability of theories; sets of sentences
11U09Connections of number theory with model theory
03C60Model-theoretic algebra