# polynomials – Extremal positivity property of complete square

Consider the bilinear form $$displaystyle A(x,y) = sum_{i, j} a_{i j} x_i y_j$$ and biquadratic form $$displaystyle B(x,y) = sum_{i, j, k, l} b_{i j k l} x_i y_j x_k y_l$$ where $$x, y in mathbb{R}^n$$ and all the coefficients are real.

Assume $$B(x,y) geq 0$$ for all $$x,y in mathbb{R}^n$$ and $$B$$ is not identically zero. I am trying to prove extremal positivity property of $$A^2(x,y)$$, namely that if $$A^2(x,y) – B(x,y) geq 0, qquad forall x, y in mathbb{R}^n$$

then $$B$$ must be a scalar multiple of $$A^2$$.