I have a sort of binary quadratic form $Q(x,y)=\lambda_1x^4 + \lambda_2 x^2y^2 +\lambda_3 y^4$. I can assume $\lambda_1, \lambda_3 > 0$. Consider $$\int dxdy \; e^{-Q(x,y)}$$ What minimal conditions on $\lambda_2$ ensure that the integral converges? I guess I need the conditions on $\lambda_2$ which ensure that Q will be positive 'everywhere at infinity' in the plane.
Apologies if I have miss-tagged the question or used inaccurate wording (I'm not a professional mathematician).
When $\lambda_2 > -2 \sqrt {\lambda_1 \lambda_3}$ the integral converges. This includes $\lambda_2 > 0.$ When $\lambda_2 < -2 \sqrt {\lambda_1 \lambda_3}$ the integral diverges.
Better to write this way: $$ A^4 x^4 + B x^2 y^2 + C^4 y^4. $$ IF $$ B > -2 A^2 C^2, $$ $$ B = D^4 -2 A^2 C^2, $$ then $$ A^4 x^4 + B x^2 y^2 + C^4 y^4 = \left( A^2 x^2 - C^2 y^2 \right)^2 + D^4 x^2 y^2 $$
Not sure yet about $\lambda_2 = -2 \sqrt {\lambda_1 \lambda_3}.$ Note $$ \left( \sqrt \lambda_1 x^2 - \sqrt \lambda_3 y^2 \right)^2 = \lambda_1 x^4 - 2 \sqrt{\lambda_1 \lambda_3} x^2 y^2 + \lambda_3 y^4, $$ so this is exactly zero along a line $y=mx$ for a fixed $m.$
Trying switching to polar coordinates.