We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$, that is, the maps from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ of the form $$( x , y ) \mapsto a x ^ { 2 } + b x y + c y ^ { 2 } + d x + e y + f \quad \text { where } \quad ( a , b , c , d , e , f ) \in \mathbb { R } ^ { 6 }$$ A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ Determine the functions in $\mathcal { P } _ { 2 }$ satisfying (1) on $\mathbb { R } ^ { 2 }$.
We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$, that is, the maps from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ of the form
$$( x , y ) \mapsto a x ^ { 2 } + b x y + c y ^ { 2 } + d x + e y + f \quad \text { where } \quad ( a , b , c , d , e , f ) \in \mathbb { R } ^ { 6 }$$
A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) if and only if
$$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$
Determine the functions in $\mathcal { P } _ { 2 }$ satisfying (1) on $\mathbb { R } ^ { 2 }$.