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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,

  1. ($q$ 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 $y$
  2. ($f$ is the quadratic form associated to the bilinear form $g$.) That is, $q(x,x)=f(x)$
  3. ($g$ 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 $S^{n-1}$.

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Alas, no. Take $q(x,y)=(x_1y_2-x_2y_1)^2$. Then $f=0$. –  fedja Jun 2 '11 at 19:06

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