Continuing a line of investigation that has studied the
function classes #P, #SAC^1, #L, and #NC^1, we study the
class of functions #AC^0. One way to define #AC^0 is as the
class of functions computed by constant-depth polynomial-size
arithmetic circuits of unbounded fan-in addition and
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In this paper we give the first deterministic polynomial time algorithm for testing whether a {\em diagonal} depth-$3$ circuit $C(\arg{x}{n})$ (i.e. $C$ is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only ... more >>>

Let $\F\{x_1,x_2,\cdots,x_n\}$ be the noncommutative polynomial
ring over a field $\F$, where the $x_i$'s are free noncommuting
formal variables. Given a finite automaton $\A$ with the $x_i$'s as
alphabet, we can define polynomials $\f( mod A)$ and $\f(div A)$
obtained by natural operations that we call ... more >>>

A folklore result in arithmetic complexity shows that the number of multiplications required to compute some $n$-variate polynomial of degree $d$ is $\sqrt{{n+d \choose n}}$. We complement this by an almost matching upper bound, showing that any $n$-variate polynomial of degree $d$ over any field can be computed with only ... more >>>