The arithmetic Jacobian matrix and determinant

Let α1,…, α_m be such real numbers that can be expressed as a finite product of prime powers with rational exponents. Using arithmetic partial derivatives, we define the arithmetic Jacobian matrix Ja of the vector a = (α₁,…, α_m) analogously to the Jacobian matrix Jf of a vector function f. We introduce the concept of multiplicative independence of {α₁,…, α_m} and show that Ja plays in it a similar role as J_f does in functional independence. We also present a kind of arithmetic implicit function 1 theorem and show that Ja applies to it somewhat analogouslytheorem and show that Ja applies to it somewhat analogously as Jf applies to the ordinary implicit function theorem.