Ruling out the existence of a strange polynomial II

This is a refinement of my question asked earlier, which is answered beautifully in the negative by Thomas Browning. The example he gave was geometrically reducible. Now I want to ask the same question, but with the extra assumptions that $$f(0,0) = 0$$ and $$f$$ is geometrically irreducible for all $$c \in \mathbb{R}$$.

For convenience, the question is: does there exist a polynomial $$f \in \mathbb{Z}[x,y]$$ such that $$f(0,0) = 0$$ and $$f$$ is geometrically irreducible, and such that

$$\displaystyle f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$

and

$$\displaystyle \liminf_{\mathbb{R}^2} f(x,y) = -\infty?$$

• How about taking that example and adding a random positive number? Nov 24 at 22:39
• Yes, I want to avoid examples like $f(x,y) = x^2 - y^2 - 2$ and such, and hoping that the assumption $f(0,0) = 0$ would eliminate such trivialities. If this is not enough then we may further assume that $f(x,y) + c$ is geometrically irreducible for all $c \in \mathbb{R}$. Nov 24 at 22:44

Just a slight modification of the previous example: $$f(x,y):=\left(5 y^2+5 y+1\right) \left(x^2+y^2\right)+\left(10 y^2+10 y+1\right) y^2.$$
Your conditions $$f(0,0) = 0$$ and $$f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$ contradict each other. So, I was assuming that you meant $$f(0,0) = 0$$ and $$f(a,b) > 0 \text{ for all } (a,b) \in \mathbb{Z}^2\setminus\{(0,0)\}.$$