Monomial-size vs. Bit-complexity in Sums-of-Squares and Polynomial Calculus

TL;DR

This paper investigates the relationship between monomial-size and bit-complexity in Sum-of-Squares and Polynomial Calculus Resolution over rationals, showing that small monomial-size does not imply low bit-complexity for certain polynomial constraints.

Contribution

It demonstrates the existence of polynomial constraints with small monomial-size refutations that require exponential bit-complexity, highlighting a separation between these complexity measures.

Findings

Small monomial-size refutations exist for certain constraints.

Exponential bit-complexity can be necessary despite low-degree refutations.

Monomial-size and bit-complexity are fundamentally different complexity measures.

Abstract

In this paper we consider the relationship between monomial-size and bit-complexity in Sums-of-Squares (SOS) in Polynomial Calculus Resolution over rationals (PCR/$\mathbb{Q}$). We show that there is a set of polynomial constraints $Q_n$ over Boolean variables that has both SOS and PCR/$\mathbb{Q}$ refutations of degree 2 and thus with only polynomially many monomials, but for which any SOS or PCR/$\mathbb{Q}$ refutation must have exponential bit-complexity, when the rational coefficients are represented with their reduced fractions written in binary.