Revisiting a recent result of Eisenträger and Everest who proved that Hilbert's tenth problem is undecidable over certain subrings of Q, two additional theorems are proved. The theorems show that we can specify certain conditions for the sets of primes which define these rings. Thus, the freedom we have when choosing these rings is further illustrated.Mathematics Subject Classification (2010). 11G05, 11U05.
We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It is not a counter-example to the resolution in positive characteristic since larger centers are also permissible and prevent the phenomenon from happening.MSC-2000: 14B05, 14E15, 12D10. We are grateful to an anonymous referee for valuable suggestions improving the presentation of the article.
In contrast to the characteristic zero situation, the residual order of an ideal may increase in positive characteristic under permissible blowups at points of the exceptional divisor where the order of the ideal has remained constant. The specific situations where this happens are described explicitly.
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