Y-compatible and strict Y-compatible functions Original Research Article

Let Y ∈ Rn. A function f : Rn → Rk is Y-compatible, if for any Z ∈ Rn, Z ≤ Y if and only if f(Z) ≤ f(Y) and is strictY-compatible, if for any Z ∈ Rn, Z < Y if and only if f(Z) < f(Y). It is proved that for any Y ∈ Rn, n ≥ 2, there is no Y-compatible polynomial function f : Rn → Rk, 1 ≤ k < n. It is also proved that for a differentiable strict Y-compatible map f, Jf(Y) = 0, where Jf(Y) denote the Jacobian matrix of the mapping f in Y. These problems arose in studying data compression of analog signatures.