The Matsaev (1971) conjecture is false for every p ≠ 2, and other miscellanea

The Matsaev (1971) conjecture asserts that for 1 < p < ∞, every contraction T on an Lp space and every polynomial P satisfy ‖P(T)‖ ≤ ‖P(S)‖, where S is the shift on ℓp(ℕ). We prove that it fails for every p ∈ (1, ∞) ∖ {2}, already for a real 2×2 matrix and a real quadratic, without numerical computation, and that it holds for every quadratic whose constant and leading coefficients have equal modulus; these results and their proofs also form the short paper "The Matsaev (1971) conjecture is false for every p ≠ 2". This expanded version adds an explicit quadratic and an explicit family of scaled rotations that violate the conjecture on a whole punctured neighbourhood of p = 2, with a proof by hand; an exact rational re-proof of Drury's counterexample at p = 4; upper bounds, certified in exact rational arithmetic, for the shift norms of Drury's polynomial at p = 3, 4, 6, …, 20; explicit counterexamples, certified by computer, for every p ∈ [40/39, 40], p ≠ 2; and conditions that every 2×2 counterexample must satisfy. The results of the short paper have also been proved in the Lean 4 proof assistant.