The existential theory of real hyperelliptic function fields (Q5925793)

The scope of this paper is to prove that the analogue of Hilbert's Tenth Problem has a negative answer for a class of real hyperelliptic function fields over real closed fields. More precisely, the author proves the following main theorem: Let \(C\) be a hyperelliptic curve, defined over a real closed field \(k\), and let the affine part of \(C\) be defined by an equation \(d^2=g(t)\), where \(g(t)\) is a polynomial which has a zero in \(k\), so that the curve \(C\) has infinitely many \(k\)-rational points. The function field \(k(C)\) has an undecidable existential theory in the language \(L(C)=\{0, 1,t,d,+,.\}\).