On the complexity of linear programming in the Blum–Shub–Smale model

We consider the Blum–Shub–Smale model of computation over the reals. It was shown that the Linear Programming Feasibility problem (LP, LPyes) (i.e., given A ∈ R^{m × n}, b ∈ R^m, does there exist an x ∈ R⁺ⁿ s.t. A × x = b?) is reducible in polynomial time to F², F²zero,+) (i.e., given a polynomial f with real coefficients and degree at most 2, does there exist a nonnegative real zero?). We show that (LP, LPyes) is polynomially equivalent to the more special decision problem F²⁺, F²⁺zero,+) (i.e., given a polynomial f ∈ F² with f(x) ⩾ 0 for all x ∈ Rⁿ, does there exist a nonnegative real zero?).