A Dual Polynomial for OR

TL;DR

This paper introduces a dual polynomial approach to prove the approximate degree lower bound of the OR function, providing a new proof technique via duality theory and linear programming.

Contribution

It constructs a dual polynomial to establish the Omega(sqrt(n)) lower bound on the approximate degree of OR, offering a novel proof method.

Findings

Proves the Omega(sqrt(n)) lower bound for OR's approximate degree

Introduces a dual polynomial construction method

Utilizes linear programming duality in complexity theory

Abstract

We reprove that the approximate degree of the OR function on n bits is Omega(sqrt(n)). We consider a linear program which is feasible if and only if there is an approximate polynomial for a given function, and apply the duality theory. The duality theory says that the primal program has no solution if and only if its dual has a solution. Therefore one can prove the nonexistence of an approximate polynomial by exhibiting a dual solution, coined the dual polynomial. We construct such a polynomial.