On polynomial submersions of degree 4 and the real Jacobian conjecture in R2R2

We prove the following version of the real Jacobian conjecture: “Let F=(p,q):R2→R2F=(p,q):R2→R2 be a polynomial map with nowhere zero Jacobian determinant. If the degree of p is less than or equal to 4, then F   is injective”. The approach to prove this result leads to a complete classification, up to affine change of coordinates, of the polynomial submersions of degree 4 in R2R2 whose level sets are not all connected.