Continuous sums of squares of forms

We give a continuous representation of positive semidefinite (psd) n-ary quadratic forms over an ordered field as sums of (almost n!e) nonnegatively-weighted squares of linear forms. This answers a question of Kreisel, who noticed in 1980 that (already for n=2) the usual “completion-of-square” process gives a discontinuous representation. For n=2 J.F. Adams has recently reduced the required number of continuous summands to 2, but only over Euclidean ordered fields. We also show that any universal representation of psd quadratic forms as sums of squares of quadratic forms must be discontinuous at (X2 +Y2)2. © 1982, North-Holland Publishing Company, Amsterda