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We generalize recent linear lower bounds for Polynomial Calculus based on binomial ideals. We produce a general hardness criterion (that we call immunity) which is satisfied by a random function and prove linear lower bounds on the degree of PC refutations for a wide class of tautologies based on immune functions. As some applications of our techniques, we introduce modp Tseitin tautologies in the Boolean case (e.g. in the presence of axioms xi2=xi), prove that they are hard for PC over fields with characteristic different from p, and generalize them to Flow tautologies which are based on the MAJORITY function and are proved to be hard over any field. We also show the Ω(n) lower bound for random k-CNFs over fields of characteristic 2.