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

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