New Lower Bounds against Homogeneous Non-Commutative Circuits

TL;DR

This paper establishes new lower bounds on the size of homogeneous non-commutative circuits for specific polynomials, advancing understanding of their computational complexity in algebraic circuit theory.

Contribution

It introduces explicit polynomials requiring large homogeneous non-commutative circuits and improves lower bounds for various polynomial classes.

Findings

Explicit polynomial requires size Ω(d/log d) circuits.

For n-variable polynomials, lower bounds are Ω(nd) or Ω(nd log n / log d).

Quadratic lower bound for ordered central symmetric polynomial.

Abstract

We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree $d$ which requires homogeneous non-commutative circuit of size $\Omega(d/\log d)$. For an $n$-variate polynomial with $n>1$, the result can be improved to $\Omega(nd)$, if $d\leq n$, or $\Omega(nd \frac{\log n}{\log d})$, if $d\geq n$. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.