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The Arithmetic Jacobian Matrix and Determinant
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Pentti Haukkanen and Jorma K. Merikoski

Faculty of Natural Sciences

FI-33014 University of Tampere

Finland

Mika Mattila

Department of Mathematics

Tampere University of Technology

PO Box 553

FI-33101 Tampere

Finland

Timo Tossavainen

Department of Arts, Communication and Education

Lulea University of Technology

SE-97187 Lulea

Sweden

**Abstract:**

Let *a*_{1},..., *a*_{m} be 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 **J**_{a} of the
vector **a** = (*a*_{1},..., *a*_{m}) analogously to the
Jacobian matrix **J**_{f} of a vector function **f**.
We introduce the concept of
multiplicative independence of {*a*_{1},..., *a*_{m}}
and show that **J**_{a} plays in it a similar role as **J**_{f} does in functional
independence. We also present a kind of arithmetic implicit function
theorem and show that **J**_{a} applies to it somewhat analogously as **J**_{f}
applies to the ordinary implicit function theorem.

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(Concerned with sequences
A000040
A003415.)

Received January 27 2017; revised versions received July 11 2017; August 1 2017.
Published in *Journal of Integer Sequences*, September 8 2017.

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