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

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.