Geodesic
A new technique for obtaining Diophantine representations via elimination of bounded universal quantifiers
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IX, Tome 220 (1995), pp. 83-92 
Yu. V. Matiyasevich. A new technique for obtaining Diophantine representations via elimination of bounded universal quantifiers. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IX, Tome 220 (1995), pp. 83-92. http://geodesic.mathdoc.fr/item/ZNSL_1995_220_a5/
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     title = {A new technique for obtaining {Diophantine} representations via elimination of bounded universal quantifiers},
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     year = {1995},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_1995_220_a5/}
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M. Davis proved in the early 1950s that every recursively enumerable set has an arithmetic representation with a unique bounded universal quantifier, known today as the Davis normal form. Davis, H. Putnam, and J. Robinson showed in 1961 how the Davis normal form can be transformed into a purely existential exponential Diophantine representation which uses not only addition and multiplication, but also exponentiation. The present author eliminated the exponentiation in 1970 and thus obtained the unsolvability of Hilbert's tenth problem. The paper presents a new method for transforming the Davis normal form into the exponential Diophantine representation. Bibliography: 12 titles.