Does there exist a bivariate polynomial $p \in \Bbb R[x,y]$ that is positive iff $x, y > 0$?
My motivation was originally to state multiple positivity conditions with one expression but now I'm just curious and unable to find the answer via search.
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Sign up to join this communityDoes there exist a bivariate polynomial $p \in \Bbb R[x,y]$ that is positive iff $x, y > 0$?
My motivation was originally to state multiple positivity conditions with one expression but now I'm just curious and unable to find the answer via search.
No.
Consider such a polynomial $P$.
$P$ must be $0$ on the positive real $x$ and $y$ axes, as it must cross from being positive to negative there. As a result, both $x$ and $y$ divide such a polynomial. Write $P=xyQ$.
Now, $Q$ is positive everywhere except the third quadrant, and so $R(x,y)=-Q(-x,-y)$ is positive only on the first quadrant. As a result, we have a polynomial $R$ that satisfies the same property, but has lower degree than $P$. This process cannot continue infinitely, giving a contradiction.