Polynomial Calculus sizes over the Boolean and Fourier bases are incomparable

Abstract: For every $n >0$, we show the existence of a CNF tautology over $O(n^2)$ variables of width $O(\log n)$ such that it has a Polynomial Calculus Resolution refutation over ${0,1}$ variables of size $O(n^3polylog(n))$ but any Polynomial Calculus refutation over ${+1,-1}$ variables requires size $2^{\Omega(n)}$. This shows that Polynomial Calculus sizes over the ${0,1}$ and ${+1,-1}$ bases are incomparable (since Tseitin tautologies show a separation in the other direction) and answers an open problem posed by Sokolov [Sok20] and Razborov.