The arithmetic Jacobian matrix and determinant

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    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 languageEnglish
    Article number17.9.2
    JournalJournal of Integer Sequences
    Issue number9
    Publication statusPublished - 2017
    Publication typeA1 Journal article-refereed


    • Arithmetic derivative
    • Arithmetic partial derivative
    • Implicit function theorem
    • Jacobian determinant
    • Jacobian matrix
    • Multiplicative independence
    • mathematics

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