Every Ternary Quintic is a Sum of Ten Fifth Powers

To our knowledge at the time of writing, the maximum Waring rank for the set of all ternary forms of degree d (with coefficients in an algebraically closed field of characteristic zero) is known only for d <= 4, and the best upper bound that is known for d=5 is twelve. In this work we lower the upper bound to ten (and there are evidences that it is likely sharp).