Automating Algebraic Proof Systems Is NP-Hard

We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula F, it is NP-hard to find a refutation of F in the Nullstellensatz, Polynomial Calculus, or Sherali-Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a simplified proof of, the recent breakthrough of Atserias and Muller (JACM 2020) that established an analogous result for Resolution.