Further results on Hilbert’s Tenth Problem

  title={Further results on Hilbert’s Tenth Problem},
  author={Zhi-Wei Sun},
  journal={Science China Mathematics},
  • Zhi-Wei Sun
  • Published 12 April 2017
  • Mathematics
  • Science China Mathematics
Hilbert’s Tenth Problem (HTP) asked for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring ℤ of integers. This was finally solved by Matiyasevich negatively in 1970. In this paper we obtain some further results on HTP over ℤ. We show that there is no algorithm to determine for any P ( z 1 ,…, z 9 ) ∈ ℤ[ z 1 ,…, z g ] whether the equation P ( z 1 ,…, z g ) = 0 has integral solutions with z 9 ⩾ 0. Consequently, there is… 
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