Let

$$P(X,Y)= c_{22}X^2Y^2 +c_{21}X^2Y +c_{12}XY^2 +c_{20}X^2 +c_{11}XY+c_{02}Y^2+c_{10}X+c_{01}Y+c_{00}$$

be a polynomial of two variables $X$ and $Y$ with real coefficients $c_{ij}$.

What are the necessary and sufficient conditions on the coefficients $c_{ij}$ such that

$P(X,Y) \geq 0$ for all pairs $(X,Y)$ with $X\geq 0$, $Y\geq 0$?

Let

$$P(X,Y)= c_{22}X^2Y^2 +c_{21}X^2Y +c_{12}XY^2 +c_{20}X^2 +c_{11}XY+c_{02}Y^2+c_{10}X+c_{01}Y+c_{00}$$

be a polynomial of two variables $X$ and $Y$ with real coefficients $c_{ij}$.

What are the necessary and sufficient conditions on the coefficients $c_{ij}$ such that $P(X,Y) \geq 0$ for all pairs $(X,Y)$ with $X\geq 0$, $Y\geq 0$?

Let

P(X,Y)= c22X^2Y^2+c21X^2Y+c12XY^2+c20X^2+c11XY+c02Y^2+c10X+c01Y+c00

be a polynomial of two variables X and Y with real coefficients cij.

What are the necessary and sufficient conditions on the coefficients cij such that

P(X,Y)>=0 for all pairs (X,Y) with X>=0, Y>=0?

Let

$$P(X,Y)= c_{22}X^2Y^2 +c_{21}X^2Y +c_{12}XY^2 +c_{20}X^2 +c_{11}XY+c_{02}Y^2+c_{10}X+c_{01}Y+c_{00}$$

be a polynomial of two variables $X$ and $Y$ with real coefficients $c_{ij}$.

What are the necessary and sufficient conditions on the coefficients $c_{ij}$ such that

$P(X,Y) \geq 0$ for all pairs $(X,Y)$ with $X\geq 0$, $Y\geq 0$?