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 Zap. Nauchn. Sem. POMI, 1995, Volume 220, Pages 83–92 (Mi znsl4282)

A new technique for obtaining Diophantine representations via elimination of bounded universal quantifiers

Yu. V. Matiyasevich

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: 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.

Full text: PDF file (404 kB)

English version:
Journal of Mathematical Sciences (New York), 1997, 87:1, 3228–3233

Bibliographic databases:

UDC: 510.57+511.53

Citation: Yu. V. Matiyasevich, “A new technique for obtaining Diophantine representations via elimination of bounded universal quantifiers”, Studies in constructive mathematics and mathematical logic. Part IX, Zap. Nauchn. Sem. POMI, 220, POMI, St. Petersburg, 1995, 83–92; J. Math. Sci. (New York), 87:1 (1997), 3228–3233

Citation in format AMSBIB
\Bibitem{Mat95} \by Yu.~V.~Matiyasevich \paper A new technique for obtaining Diophantine representations via elimination of bounded universal quantifiers \inbook Studies in constructive mathematics and mathematical logic. Part~IX \serial Zap. Nauchn. Sem. POMI \yr 1995 \vol 220 \pages 83--92 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl4282} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1374097} \zmath{https://zbmath.org/?q=an:0940.03052} \transl \jour J. Math. Sci. (New York) \yr 1997 \vol 87 \issue 1 \pages 3228--3233 \crossref{https://doi.org/10.1007/BF02358996}