Let $p=p(x,y) \in \mathbb{R}[x,y]$ be a two variable polynomial over $\mathbb{R}$. Assume that:
(1) All monomials in $p$ are of (total) even degrees, namely, $p$ is of the form: $p=p_{00}+p_{20}x^2+p_{11}xy+p_{02}y^2+\cdots$, where $p_{ij} \in \mathbb{R}$.
(2) For all $x,y \in \mathbb{R}$, $p(x,y) \geq 0$.
Is it possible to say something interesting about the coefficients of such $p$?
I guess that there should be some relation between the coefficients $p_{ij}$ and the degrees of the monomials, though it seems quite complicated to find it.
Remark: This question is relevant.
Edit: What if the $p_{ij}$'s belong to one of the following four sets: $\{1,-1,0\}$; $\mathbb{Z}$; $\mathbb{N}$; $\mathbb{R}^{+}$? For example: In the special case $p=a+bx^2+cxy+dy^2$, with $a,b,c,d \in \mathbb{R}^{+}$, if I am not wrong, such $p$ will be positive if $c< min \{b,d\}$ and $b-c+d \geq 0$ (the second condition can be replaced with $a>|b-c+d|$).
Perhaps (if I am not missing something) the situation is similar in higher degrees, for example: $p=a+bx^2+cxy+dy^2+ex^4+fx^3y+gx^2y^2+hxy^3+iy^4$, with all the coefficients in $\mathbb{R}^{+}$ will be positive if $c < \min \{e,g,i\}$, $\max \{f,h \} < \min \{e,g,i\}$, $b-c+d \geq 0$ and $e-f+g-h+i \geq 0$ (the third and fourth conditions can be replaced by $a > 2 \max\{|b-c+d|, |e-f+g-h+i|\}$).
It seems that similar arguments will be valid for a more general case, in which only the highest degree $m$ is required to be even, without restriction on the parity of lower degrees (but still under the assumption that all the coefficients belong to $\mathbb{R}^{+}$, of course).
Any hints and comments are welcome!
Edit: This paper more or less answers my question, by applying Proposition 1.15 to each homogeneous component of my given polynomial $p$. If there are $r$ homogeneous components in $p$, then we have $r$ conditions, if all satisfied, then $p$ is positive. (Hankel quadratic form, mentioned in its last section, also sounds interesting). However, the ideas in that paper are new to me, so any help in showing how to apply them in practice is welcome.