by  S.Yu. Orevkov   and   E. I. Shustin

Abstract:

We show that there exists a real non-singular pseudoholomorphic sextic curve in the affine plane which is not isotopic to any real algebraic sextic curve. This result completes the isotopy classification of real algebraic affine M-curves of degree 6. Comparing with the isotopy classification of real affine pseudoholomorphic sextic M-curve, obtained earlier by the first author, one obtains three pseudoholomorphic isotopy types which are algebraically unrealizable. In a similar way we find a real pseudoholomorphic, algebraically unrealizable (M-1)-curve of degree 8 on a quadratic cone with a special position with respect to a generating line. The proofs are based on the Hilbert-Rohn-Gudkov approach developed by the second author and the cubic resolvent method developed by the first author.