On vanishing sums of roots of unity in polynomial calculus and sum-of-squares

Abstract

We introduce a novel take on sum-of-squares that is ableto reason with complex numbers and still make use of polynomial inequalities.This proof system might be of independent interest since itallows to represent multivalued domains both with Boolean and Fourierencoding. We show degree and size lower bounds in this system for anatural generalization of knapsack: the vanishing sums of roots of unity.These lower bounds naturally apply to polynomial calculus as-well.