2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/104845The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools to prove the symmetric Jacobian Conjecture for the case $F=X-H$ with $H$ homogeneous and $JH^{3}=0$. Other special results are also derived. We pose a combinatorial statement which would give a complete proof the Jacobian Conjecture.19 pages; submitted for publication in an upcoming volume honoring Masayoshi MiyanishiCombinatoricsCommutative AlgebraAlgebraic Geometry14R15; 05C05; 13A99text