Sum of Squares (SOS)

f = ∑ g_i^2 h_i ≥ 0 — the universal hammer for symmetric inequalities.

The idea

The sum of squares method proves an inequality by writing it as a statement that squares are nonnegative.

Move everything to one side, so that the claim reads $f \ge 0$, and then exhibit a certificate.

Definition (Sum-of-squares certificate).

A sum-of-squares certificate for $f \ge 0$ is an identity $f \;=\; \sum_i g_i^{2} \, h_i, \qquad h_i \ge 0,$ in which the $g_i$ and $h_i$ are expressions in the variables and each weight $h_i$ is nonnegative on the domain in question.

Once $f$ is in this shape there is nothing left to argue: every term is a nonnegative weight times a square, so the sum cannot be negative.

Such a decomposition proves the claim by being exhibited, and it can be checked by expanding, with no appeal to any other inequality. It also delivers the equality case, since the total vanishes exactly when every term does.

For symmetric expressions in three variables, the standard shape to aim for is $f = S_a(b-c)^{2} + S_b(a-c)^{2} + S_c(a-b)^{2},$ where the coefficients $S_a, S_b, S_c$ are expressions in $a, b, c$ rather than constants. The three squared differences vanish when the variables agree, which is where symmetric inequalities are usually tight, so the work reduces to checking the signs of the coefficients.

Ways to work on it

• Walkthrough. Prove an inequality by rewriting it as a sum of squares, first in two variables and then in three.
• Practice. Check a sum-of-squares identity at specific numbers and name the inequality it proves.
• Hardest. Identify the sum-of-squares decomposition of a given symmetric expression.

Not sure where to start? Take the ten-question placement test.