We work in the multivariate case so that $x$ stands for $(x_1,\ldots, x_n)$. Let $q(x,y)$ be a symmetric matrix representation of a homogeneous polynomial $f(x)$ of degree $d$. Explicitly,
- ($q$ is a bilinear form in the monomials of degree $d$. )That is, $q(x,y)$ is a homogeneous of degree $d$ in $x$ and also of degree $d$ in $y$
- ($f$ is the quadratic form associated to the bilinear form $g$.) That is, $q(x,x)=f(x)$
- ($g$ is symmetric.) That is, $g(x,y)=g(x,y).$
If we write $g(x,y)=\sum_{|\alpha|=|\beta|=d} a_{\alpha\beta}x^\alpha b^\beta$, then condition 3 means that $a_{\alpha\beta}=a_{\beta\alpha}$.
Suppose that $f$ is positive away from the origin. Is there an estimate of $|q|$ on the product of unit spheres $S^{n-1}\times S^{n-1}$ in terms of quantities involving $f$? For instance the supremum and infimum of $f$ on the unit sphere $S^{n-1}$.
