Abstract
Let α1,…, αm 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,…, αm) analogously to the Jacobian matrix Jf of a vector function f. We introduce the concept of multiplicative independence of {α1,…, αm} and show that Ja plays in it a similar role as Jf 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.
Original language | English |
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Article number | 17.9.2 |
Journal | Journal of Integer Sequences |
Volume | 20 |
Issue number | 9 |
Publication status | Published - 2017 |
Publication type | A1 Journal article-refereed |
Keywords
- Arithmetic derivative
- Arithmetic partial derivative
- Implicit function theorem
- Jacobian determinant
- Jacobian matrix
- Multiplicative independence
- mathematics
Publication forum classification
- Publication forum level 1