View source for Diophantine predicate

Any predicate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d0326401.png" /> defined on the set of (ordered) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d0326402.png" />-tuples of integers (or non-negative integers or positive integers) for which there exists a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d0326403.png" /> with integer coefficients such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d0326404.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d0326405.png" /> satisfies the predicate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d0326406.png" /> if and only if the Diophantine equation (cf. [[Diophantine equations|Diophantine equations]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d0326407.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

is solvable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d0326408.png" />. The truth domain of a Diophantine predicate is a [[Diophantine set|Diophantine set]]. The class of Diophantine predicates coincides with the class of recursively enumerable predicates (cf. [[Diophantine equations, solvability problem of|Diophantine equations, solvability problem of]]).



====Comments====
The truth domain of a Diophantine predicate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d0326409.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d03264010.png" />-tuples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d03264011.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d03264012.png" />, i.e., for which (*) is solvable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032640/d03264013.png" />.