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 $X\geq 0$, $Y\geq 0$?**



Nonnegativity conditions for a polynomial in two variables?
There is [Stengle's positivensatz,](http://en.wikipedia.org/wiki/Stengle%27s_Positivstellensatz), otherwise, if your polynomial were homogeneous, you could use Polya's theorem, and if you could make your domain compact, you can use Handelman's theorem, but otherwise, Stengle is your man. 


For Polya and Handelman see Powers and Reznick:



```
\bib{MR1854339}{article}{
  author={Powers, Victoria},
  author={Reznick, Bruce},
  title={A new bound for P\'olya's theorem with applications to polynomials
  positive on polyhedra},
  note={Effective methods in algebraic geometry (Bath, 2000)},
  journal={J. Pure Appl. Algebra},
  volume={164},
  date={2001},
  number={1-2},
  pages={221--229},
  issn={0022-4049},
  review={\MR{1854339 (2002g:14087)}},
  doi={10.1016/S0022-4049(00)00155-9},

```

}


