Pythagoras numbers for ternary forms

We study the Pythagoras numbers py ⁡ ( 3 , 2 d ) py(3,2d) of real ternary forms, defined for each degree 2 d 2d as the minimal number r r such that every degree 2 d 2d ternary form which is a sum of squares can be written as the sum of at most r r squares of degree d d forms. Scheiderer [Math. Z. 286 (2017), pp. 559–570] showed that d + 1 ≤ py ⁡ ( 3 , 2 d ) ≤ d + 2 d+1≤ py(3,2d)≤ d+2 . We show that p y ( 3 , 2 d ) = d + 1 py(3,2d) = d+1 for 2 d = 8 , 10 , 12 2d = 8,10,12 . Our approach is to connect the problem with optimality conditions for a Euclidean distance problem. When paired with Diesel’s characterization of height 3 Gorenstein algebras, this allows us to control the syzygies of the forms involved in the decomposition into sums of squares.