In this paper we show that the rings of regular functions on two real algebraic curves over the same real closed field are Witt equivalent (i.e. their Witt rings are isomorphic) if and only if the curves have the same number of semi-algebraically connected components. Moreover, in the second part of the paper, we prove that every strong isomorphism of Witt rings of rings of regular functions can be extended to an isomorphism of Witt rings of fields of rational functions. This extension is not unique, though.
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