  We introduce a system of algebraic invariants of isotopy classes of Morse
polynomials ${\mathbb R}^2 \to {\mathbb R}^1$ of degree $d$ with non-degenerate
principal homogeneous part. In the first non-trivial case of $d=4$ (in which
these invariants are hypothetically complete) we calculate all 71 possible
values of these invariants and realize all these values by concrete Morse
polynomials.
