Pseudoholomorphic algebraically unrealizable curves

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 this with the isotopy classification of real affine pseudoholomorphic sextic $M$-curves obtained earlier by the first author, we find 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 arranged in a special way with respect to a generating line. The proofs are based on the Hilbert–Rohn–Gudkov approach developed by the second author and on the cubic resolvent method developed by the first author.