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

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/Q). We show that there is a set of polynomial constraints Q n over Boolean variables that has both SOS and PCR/Q refutations of degree 2 and thus with only polynomially many monomials, but for which any SOS or PCR/Q refutation must have exponential bit-complexity, when the rational coefficients are represented with their reduced fractions written in binary.