# Arithmetic Circuit Hierarchy?

The answers to the following question -

Hierarchy theorem for circuit size

give a "circuit hierarchy theorem" for boolean circuits. Does there exist a similar hierarchy theorem for arithmetic circuits? In particular, for $$f(n) >> g(n)$$ (for some notion of >>) does there exist a family of polynomials $$h(x_1, \cdots, x_n) \in \mathbb{F}(x_1, \cdots x_n)$$, $$n \geq 1$$, computable by arithmetic circuits of size $$f(n)$$ but not by arithmetic circuits of size $$g(n)$$?

• I think an idea similar to the second answer works. May 21, 2023 at 3:59

This question has a somewhat trivial answer because the polynomial $$x^{2^s}$$ requires $$s$$ multiplications, so you can just take $$h = x_1^{2^{f(n)}}$$. This is one of the reasons why in algebraic complexity we almost always talk about families with polynomially bounded degree.
Main idea: in a circuit computing a non-constant function has $$s$$ gates, then it has at most $$s$$ constants. By fixing the structure of the circuit and varying the constants we see that the set of all polynomials computed by the fixed circuit (which is a Zariski constructible set by Chevalley's theorem) has dimension at most $$s$$. There is only finite number of circuits of complexity $$s$$, so the set of all polynomials with complexity $$s$$ has dimension at most $$s$$. Comparing $$s$$ with the dimension of the set of polynomials with given degree and number of variables, we can get a statement about the existence of hard polynomials.