NON-DIOPHANTINE SETS IN RINGS OF FUNCTIONS

Abstract Except for a limited number of cases, a complete classification of the Diophantine (i.e., positive existentially definable) sets of polynomial rings and fields of rational functions seems out of reach at present. We contribute to this problem by proving that several natural sets and relations over these structures are not Diophantine. For instance, we show that the relation of equality of degrees is not Diophantine over the field of complex rational functions in one variable and, in the same structure, we show that certain family of relations that approximates the valuation ring at infinity is not Diophantine either.