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Let<jats:italic>M\/K<\/jats:italic>be a finite extension and let<jats:italic>W<\/jats:italic>be a set of primes of<jats:italic>K<\/jats:italic>such that all but finitely many primes of<jats:italic>W<\/jats:italic>do not split in the extension<jats:italic>M\/K<\/jats:italic>. Then there exists a set<jats:italic>W\u2032<\/jats:italic>of<jats:italic>K<\/jats:italic>-primes such that Hilbert's Tenth Problem is not decidable over<jats:italic>O<\/jats:italic><jats:sub><jats:italic>K,W\u2032<\/jats:italic><\/jats:sub>= {<jats:italic>x<\/jats:italic>\u03f5<jats:italic>K<\/jats:italic>\u2223ord<jats:sub>\u00fe<\/jats:sub><jats:italic>x<\/jats:italic>\u2265 0, \u2200\u00fe \u2209<jats:italic>W\u2032<\/jats:italic>}, and the set (<jats:italic>W<\/jats:italic>\u2032 \u2216<jats:italic>W<\/jats:italic>)\u222a{<jats:italic>W<\/jats:italic>\u2216<jats:italic>W<\/jats:italic>\u2032) is finite.<\/jats:p><jats:p>Let<jats:italic>K<\/jats:italic>be a function field of one variable over a constant field<jats:italic>C<\/jats:italic>finitely generated over \u211a. Let<jats:italic>M\/K<\/jats:italic>be a finite extension and let<jats:italic>W<\/jats:italic>be a set of primes of<jats:italic>K<\/jats:italic>such that all but finitely many primes of<jats:italic>W<\/jats:italic>do not split in the extension<jats:italic>M\/K<\/jats:italic>and the degree of all the primes in<jats:italic>W<\/jats:italic>is bounded by<jats:italic>b<\/jats:italic>\u03f5 \u2115. Then there exists a set<jats:italic>W<\/jats:italic>\u2032 of<jats:italic>K<\/jats:italic>-primes such that \u2124 has a Diophantine definition over<jats:italic>O<\/jats:italic><jats:sub><jats:italic>K,<jats:italic>W<\/jats:italic>\u2032<\/jats:italic><\/jats:sub>, and the set (<jats:italic>W<\/jats:italic>\u2032 \u2216<jats:italic>W<\/jats:italic>)\u222a(<jats:italic>W<\/jats:italic>\u2216<jats:italic>W<\/jats:italic>\u2032) is finite.<\/jats:p>","DOI":"10.2178\/jsl\/1190150109","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:13:31Z","timestamp":1197573211000},"page":"759-786","source":"Crossref","is-referenced-by-count":7,"title":["On Dipphantine definability and decidability in some rings of algebraic functions of characteristic 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