Are there polynomial time computable polynomials with circuits of size $s$ but no circuits of size $s-1$?

Question

So I was wondering whether you could always have a multivariate polynomial $P$ over $\mathbb{Z}$ that ...

• can be represented by arithmetic circuits of size $s$
• has polynomial degree and exponentially bounded coefficients (so $P$ is computable in polynomial time on some input)
• which is not computable by circuits of size $s-1$?
• is not a constant

for any $s$.

I do not need to be able to find such polynomials quickly, I guess this would be very hard task. Actually, I would just like to know if these exist.