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If I have $n$ variables $x_1, \cdots, x_n$, and a set $S$ of inequalities of the form $p(x_1, \cdots, x_n) > 0$ where $p$ is a homogenous real polynomial, is it true that I need $|S| \geq n$ to prove that $x_i > 0$ for each $i$?

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No. Take $n=3$ with the equations $x^2+y^2+z^2 < 2(xy+xz+yz)$ and $x+y+z > 0$.

The first inequality is a double cone tangent to all three coordinate planes; one component lies in the positive orthant and the other in the negative orthant.

The second inequality is a half space which makes sure we are in the first component.

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