We study in detail the special case of Waring’s problem when the power k = 2. Ultimately, we prove that four is the least number of squares needed to represent any integer. To this end, we prove that some numbers cannot be represented as sums of two squares, some cannot be represented as sums of three, and all can be represented as sums of four. We also show that numbers of a certain form can b...

Consider the problem of finding triples numbers ( x 1 , 2 3 ) and y satisfying (1) $${x_1} + {x_2} {x_3} = {y_1} {y_2} {y_3}$$ (2) $$x_1^2 x_2^2 x_3^2 y_1^2 y_2^2 y_3^2.$$

We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinit...