I have the following polynomial:
$P(a,b,c,d) = -2 a + 3 a b + 3 a^2 b + 6 a c + 6 b c + 2 d + 3 a d - 3 a^2 d - 3 b d - 6 a b d - 3 d^2 + 3 b d^2 + d^3 \;,$
where $a$, $b$, $c$, $d$ are integers $\geq 0$.
I would like to prove that $ P(a,b,c,d) \geq 0$.
I checked numerically that this is indeed the case, but I need a formal proof. Do you know how I could do this? Thank you!
For $a=0$ we have $$ P(a,b,c,d)=3((d - 1)d + 2c)b + (d - 1)(d - 2)d, $$ which is nonnegative, because we only need to check $d=0,1$, in which case the result is $6bc\ge 0$. The idea is now to proceed by induction on $a$.