On the quadratic length of plane Cremona maps of degree 4

Every non-linear plane Cremona map can be decomposed into quadratic maps, and the minimum number of quadratic maps required is called its quadratic length. It is known that plane Cremona maps of degree 3 have quadratic length either 2 or 3. In this paper, we study the quadratic length of plane Cremona maps $\phi$ of degree 4. Recall that either $\phi$ is de Jonquières, i.e. it has a base point of multiplicity 3, or it is not de Jonquières. In the latter case, $\phi$ has quadratic length 2, whereas in the former case we prove that $\phi$ has quadratic length 3, 4, or 5, and we classify those maps that reach the upper bound.