  We study a family of dissections of flow polytopes arising from the
subdivision algebra. To each dissection of a flow polytope, we associate a
polynomial, called the left-degree polynomial, which we show is invariant of
the dissection considered (proven independently by Grinberg). We prove that
left-degree polynomials encode integer points of generalized permutahedra.
Using that certain left-degree polynomials are related to Grothendieck
polynomials, we resolve special cases of conjectures by Monical, Tokcan, and
Yong regarding the saturated Newton polytope property of Grothendieck
polynomials.
