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Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:

(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;

(ii) $f$ is non-degenerate, in the sense that there isn't a non-singular linear change of variables that turns $f$ into a polynomial in one variable.

(iii) The leading homogeneous part has degree at least $4$.

Question: Does the double integral $$\iint_{\mathbb{R}^2} \frac{1}{f}$$ converge?

Comment: It is easy to see from (i) that the leading homogeneous part of $f$ must be of even degree, and cannot have in its factorization over $\mathbb{R}$ any linear factors of odd exponent.

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:

(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;

(ii) $f$ is non-degenerate, in the sense that there isn't a non-singular change of variables that turns $f$ into a function of one variable.

(iii) The leading homogeneous part has degree at least $4$.

Question: Does the double integral $$\iint_{\mathbb{R}^2} \frac{1}{f}$$ converge?

Comment: It is easy to see from (i) that the leading homogeneous part of $f$ must be of even degree, and cannot have in its factorization over $\mathbb{R}$ any linear factors of odd exponent.

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:

(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;

(ii) $f$ is non-degenerate, in the sense that there isn't a non-singular linear change of variables that turns $f$ into a polynomial in one variable.

Question: Does the double integral $$\iint_{\mathbb{R}^2} \frac{1}{f}$$ converge?

Comment: It is easy to see from (i) that the leading homogeneous part of $f$ must be of even degree, and cannot have in its factorization over $\mathbb{R}$ any linear factors of odd exponent. It is also easy to show if the degree is 2, by combining (i) and (ii), that the integral is comparable with $\iint 1/(x^2+y^2)$ and so converges. However, I can't seem to make this argument work for higher degrees.,

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:

(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;

(ii) $f$ is non-degenerate, in the sense that there isn't a non-singular linear change of variables that turns $f$ into a polynomial in one variable.

(iii) The leading homogeneous part has degree at least $4$.

Question: Does the double integral $$\iint_{\mathbb{R}^2} \frac{1}{f}$$ converge?

Comment: It is easy to see from (i) that the leading homogeneous part of $f$ must be of even degree, and cannot have in its factorization over $\mathbb{R}$ any linear factors of odd exponent.