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Given a Newton polytope $P$ inside the $d$-scaled simplex ( i.e $\alpha \in P \implies | \alpha |=d $ ), then we define the following quantity:

$$ P(x)= \left( \sum_{\alpha \in P} \binom{d}{\alpha} x^{2\alpha} \right)^{\frac{1}{2}}$$

here we use multi-index notation $\binom{d}{\alpha}=\frac{d!}{\alpha_1! \alpha_2!\ldots \alpha_n!}$ and $x^{2\alpha}=x_1^{2\alpha_1} x_2^{2\alpha_2} \ldots x_n^{2\alpha_n}$.

For instance if $P$ is the full simplex then $P(x)^2=(x_1^2+x_2^2+\ldots+x_n^2)^d$. Now assume we impose mild assumptions on $P$ that guarantees $P(x) > 0$ for all $x \in S^{n-1}$. We define

$$ A_p= \int_{S^{n-1}} \frac{1}{P(x)^{n-1}} \sigma(x) $$

where $\sigma$ is the rotation invariant measure on $S^{n-1}$ with $\sigma(S^{n-1})=1$. I need an idea to bound $A_P$ from up. I even appreciate concrete examples that you can suggest in which case we can bound $A_P$ from up. This quantity $A_P$ helps to control computationwise how nasty can a polynomial system created out of $n-1$ polynomials with support $P$ be.

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