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

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.