The Arithmetic Jacobian Matrix and Determinant

Pentti Haukkanen and Jorma K. Merikoski Faculty of Natural Sciences FI-33014 University of Tampere Finland

                pentti.haukkanen@uta.fi
                jorma.merikoski@uta.fi

Mika Mattila Department of Mathematics Tampere University of Technology PO Box 553 FI-33101 Tampere Finland

                  mika.mattila@tut.fi

Timo Tossavainen Department of Arts, Communication and Education Lulea University of Technology SE-97187 Lulea Sweden

                timo.tossavainen@ltu.se

Abstract

Let a_{1}, …, a_{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 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.

1 Introduction

Let ℝ, ℚ, ℤ, ℕ, and ℙ stand for the set of real numbers, rational numbers, integers, nonnegative integers, and primes, respectively.

If a ∈ ℝ, there may be a sequence of rational numbers (ν_{p}(a))_{p∈ℙ} with only finitely many nonzero terms satisfying

a = (sgn a) ∏_{p∈ℙ} p^{ν_{p}(a)},  (1)

where sgn is the sign function. We let ℝ′ and ℝ′_{+} denote the set of all such real numbers and the subset consisting of its positive elements, respectively. Formula (1) is also valid for a = 0, as we define ν_{p}(0) = 0 for all p ∈ ℙ. If ν_{p}(a) ≠ 0, we say that p divides a and denote p | a. Otherwise, we denote p ∤ a.

Proposition 1. Let a ∈ ℝ′ and V_{a} = {ν_{p}(a) | p ∈ ℙ}. Then

(a) a ∈ ℤ if and only if V_{a} ⊂ ℕ;

(b) a ∈ ℚ if and only if V_{a} ⊂ ℤ.

Proof. Simple and omitted.

Proposition 2. If a ∈ ℝ′, then the sequence (ν_{p}(a))_{p∈ℙ} is unique.

Proof. This is well known if a ∈ ℚ. Otherwise, see [8, Lemma 1].

We define the arithmetic derivative of a ∈ ℝ′ by

a′ = a ∑_{p∈ℙ} \frac{ν_{p}(a)}{p} = ∑_{p∈ℙ} a′_{p},

where

a′_{p} = \frac{ν_{p}(a)}{p}a  (2)

is the arithmetic partial derivative of a with respect to p. For the background and history of these concepts, see, e.g., [1, 8, 4, 3]. These references mainly concern the arithmetic derivative in ℕ, ℤ, or ℚ, but most of the results can be extended to ℝ′ in an obvious way, see [8, Section 9].

Let f = (f_{1}, …, f_{m}) : E → ℝ^{m} be a continuously differentiable function, where E ⊆ ℝ^{n} is a connected open set. Its Jacobian matrix at t = (t_{1}, …, t_{n}) ∈ E is defined by

J_{f}(t) = ( (f_{1})′_{t_{1}}(t) (f_{1})′_{t_{2}}(t) … (f_{1})′_{t_{n}}(t) (f_{2})′_{t_{1}}(t) (f_{2})′_{t_{2}}(t) … (f_{2})′_{t_{n}}(t) ⋮ (f_{m})′_{t_{1}}(t) (f_{m})′_{t_{2}}(t) … (f_{m})′_{t_{n}}(t) ) ,

where (f_{i})′_{t_{j}} = ∂f_{i}/∂t_{j}. If m = n, then det J_{f}(x) is the Jacobian determinant (or, more briefly, the Jacobian) of f.

Let a_{1}, …, a_{m} ∈ ℝ′_{+} (actually, we could study ℝ′ instead of ℝ′_{+}, which, however, would not benefit us in any significant way), and denote

P = {p_{1}, …, p_{n}} = {p ∈ ℙ | ∃a_{i} : p | a_{i}}  (3)

and

α_{ij} = ν_{p_{j}}(a_{i}), i = 1, …, m, j = 1, …, n.  (4)

Then

a_{i} = ∏_{p∈ℙ} p^{ν_{p}(a_{i})} = p_{1}^{α_{i1}}p_{2}^{α_{i2}} ⋯ p_{n}^{α_{in}}, i = 1, …, m.  (5)

Further, let

a = ( a_{1} ; a_{2} ; ⋮ ; a_{m} ) ,  α_{i} = ( α_{i1} ; α_{i2} ; ⋮ ; α_{in} ) , i = 1, …, m,  (6)

and

A_{a} = ( α_{11} α_{12} … α_{1n} ; α_{21} α_{22} … α_{2n} ; ⋮ ; α_{m1} α_{m2} … α_{mn} ) = ( α_{1}^{T} ; α_{2}^{T} ; ⋮ ; α_{m}^{T} ) .  (7)

We define the arithmetic Jacobian matrix of a by

J_{a} = ( (a_{1})′_{p_{1}} (a_{1})′_{p_{2}} … (a_{1})′_{p_{n}} ; (a_{2})′_{p_{1}} (a_{2})′_{p_{2}} … (a_{2})′_{p_{n}} ; ⋮ ; (a_{m})′_{p_{1}} (a_{m})′_{p_{2}} … (a_{m})′_{p_{n}} )

and, if m = n, the arithmetic Jacobian determinant (or, more briefly, the arithmetic Jacobian) of a by

det J_{a} = | (a_{1})′_{p_{1}} (a_{1})′_{p_{2}} … (a_{1})′_{p_{m}} ; (a_{2})′_{p_{1}} (a_{2})′_{p_{2}} … (a_{2})′_{p_{m}} ; ⋮ ; (a_{m})′_{p_{1}} (a_{m})′_{p_{2}} … (a_{m})′_{p_{m}} | .

Let f be as above. The functions f_{1}, …, f_{m} are functionally independent (i.e., there is no function ϕ : ℝ^{m} → ℝ such that ∇ϕ(f(t)) ≠ 0 and ϕ(f_{1}(t), …, f_{m}(t)) = 0 for all t ∈ E) if and only if m ≤ n and rank J_{f}(t) = m for all t ∈ E. (See, e.g., [5].) In Section 2, we will define the concept of multiplicative independence of the numbers a_{1}, …, a_{m} and study the role of J_{a} there.

We outline the implicit function theorem [7, Theorem 9.28]. Assuming m < n, write t = (x, y), where x ∈ ℝ^{m} and y ∈ ℝ^{n−m}. Let a ∈ ℝ^{m} and b ∈ ℝ^{n−m} satisfy (a, b) ∈ E. Define the function ϕ(x) = f(x, b), where x is “close to” a. If it satisfies det J_{ϕ}(a, b) ≠ 0 and if y is “close to” b, then the equation f(x, y) = 0 is uniquely “solvable” with respect to x. We will in Section 3 present a theorem where the arithmetic Jacobian matrix and determinant play a somewhat similar role. Section 4 is devoted to some concluding remarks.

2 Multiplicative independence

Let seq ℚ be the set of infinite sequences of rational numbers with finitely many nonzero terms. Consider the mapping

ν : ℝ′_{+} → seq ℚ : ν(a) = (ν_{p}(a))_{p∈ℙ}.

Defining in seq ℚ addition and scalar multiplication in the ordinary way, this set becomes a vector space over ℚ. On the other hand, defining in ℝ′_{+} addition as the ordinary multiplication and scalar multiplication as the ordinary exponentation, also ℝ′_{+} becomes a vector space over ℚ. Then ν is an isomorphism, and we can identify ℝ′_{+} with seq ℚ. Because linear independence is a well-defined concept in seq ℚ, we may so define linear independence in ℝ′_{+}. However, we find the term “multiplicative independence” more appropriate. (We quote this term from Pong [6], who studied this concept in an Abelian group.) So we say that a set

S = {a_{1}, …, a_{m}} ⊂ ℝ′_{+}  (8)

is (and the numbers a_{1}, …, a_{m} are) multiplicatively independent if the set {ν(a_{1}), …, ν(a_{m})} is linearly independent. Otherwise, S is (and a_{1}, …, a_{m} are) multiplicatively dependent.

Proposition 3. Let S be as in (8) and α_{1}, …, α_{m} as in (6). The following conditions are equivalent:

(a) The set S is multiplicatively independent.

(b) The only rational numbers λ_{1}, …, λ_{m} satisfying a_{1}^{λ_{1}} ⋯ a_{m}^{λ_{m}} = 1 are λ_{1} = ⋯ = λ_{m} = 0.

(c) The set {α_{1}, …, α_{m}} is linearly independent in the vector space ℚ^{n}, where n is as in (3).

Proof. Simple and omitted.

We now present such properties of the arithmetic Jacobian determinant that have relevance to multiplicative independence.

Proposition 4. Let a be as in (6). If n = m in (3), then

det J_{a} = \frac{a_{1}a_{2} ⋯ a_{m}}{p_{1}p_{2} ⋯ p_{m}} det A_{a}.

Proof. By (2),

det J_{a} = | \frac{α_{11}}{p_{1}} a_{1}  \frac{α_{12}}{p_{2}} a_{1} … \frac{α_{1m}}{p_{m}} a_{1} ; \frac{α_{21}}{p_{1}} a_{2}  \frac{α_{22}}{p_{2}} a_{2} … \frac{α_{2m}}{p_{m}} a_{2} ; ⋮ ; \frac{α_{m1}}{p_{1}} a_{m} \frac{α_{m2}}{p_{2}} a_{m} … \frac{α_{mm}}{p_{m}} a_{m} | .

Take the factor a_{1} from the first row, a_{2} from the second one, etc. Further, take the factor 1/p_{1} from the remaining first column, 1/p_{2} from the second one, etc. We obtain

det J_{a} = a_{1}a_{2} ⋯ a_{m} | \frac{α_{11}}{p_{1}}  \frac{α_{12}}{p_{2}} … \frac{α_{1m}}{p_{m}} ; \frac{α_{21}}{p_{1}}  \frac{α_{22}}{p_{2}} … \frac{α_{2m}}{p_{m}} ; ⋮ ; \frac{α_{m1}}{p_{1}}  \frac{α_{m2}}{p_{2}} … \frac{α_{mm}}{p_{m}} | = \frac{a_{1}a_{2} ⋯ a_{m}}{p_{1}p_{2} ⋯ p_{m}} | α_{11} α_{12} … α_{1m} ; α_{21} α_{22} … α_{2m} ; ⋮ ; α_{m1} α_{m2} … α_{mm} | .

Corollary 5. If a is as in (6), then rank J_{a} = rank A_{a}.

Proof. Given I ⊆ {1, …, m} and J ⊆ {1, …, n} with equal number of elements, let J_{a}(I, J) and A_{a}(I, J) denote the submatrix of J_{a} and respectively A_{a} with row indices in I and column indices in J. By Proposition 4, these matrices are either both nonsingular or both singular. Since rank is the largest dimension of a nonsingular square submatrix, the claim therefore follows.

Theorem 6. Let S, a, and P be as in (8), (6), and (3), respectively. The set S is multiplicatively independent if and only if rank J_{a} = m.

Proof. Apply Proposition 3 and Corollary 5. (Note that rank J_{a} = m implies m ≤ n.)

Proposition 7. Let a and P be as in (6) and (3), respectively, let 0 ≠ x_{1}, …, x_{m} ∈ ℚ, and denote a^{x} = (a_{1}^{x_{1}}, …, a_{m}^{x_{m}}). Then

rank J_{a^{x}} = rank J_{a}.

Proof. In general, we have

(a^{x})′_{p} = \frac{ν_{p}(a^{x})}{p}a^{x} = \frac{xν_{p}(a)}{p}a^{x} = xa^{x−1} \frac{ν_{p}(a)}{p}a = xa^{x−1}a′_{p}

for all a ∈ ℝ′_{+}, x ∈ ℚ, p ∈ ℙ. Consequently,

J_{a^{x}} = ( (a_{1}^{x_{1}})′_{p_{1}} (a_{1}^{x_{1}})′_{p_{2}} … (a_{1}^{x_{1}})′_{p_{n}} ; (a_{2}^{x_{2}})′_{p_{1}} (a_{2}^{x_{2}})′_{p_{2}} … (a_{2}^{x_{2}})′_{p_{n}} ; ⋮ ; (a_{m}^{x_{m}})′_{p_{1}} (a_{m}^{x_{m}})′_{p_{2}} … (a_{m}^{x_{m}})′_{p_{n}} ) =

( x_{1}a_{1}^{x_{1}−1}(a_{1})′_{p_{1}} x_{1}a_{1}^{x_{1}−1}(a_{1})′_{p_{2}} … x_{1}a_{1}^{x_{1}−1}(a_{1})′_{p_{n}} ; x_{2}a_{2}^{x_{2}−1}(a_{2})′_{p_{1}} x_{2}a_{2}^{x_{2}−1}(a_{2})′_{p_{2}} … x_{2}a_{2}^{x_{2}−1}(a_{2})′_{p_{n}} ; ⋮ ; x_{m}a_{m}^{x_{m}−1}(a_{m})′_{p_{1}} x_{m}a_{m}^{x_{m}−1}(a_{m})′_{p_{2}} … x_{m}a_{m}^{x_{m}−1}(a_{m})′_{p_{n}} ) = DJ_{a},

where

D = diag (x_{1}a_{1}^{x_{1}−1}, …, x_{m}a_{m}^{x_{m}−1}).

Since D is invertible, the claim follows.

3 An arithmetic implicit function theorem

In this section, we establish a kind of arithmetic implicit function theorem where the arithmetic Jacobian matrix and determinant apply. The problem is that arithmetic differentiation operates on numbers, not on functions; so, we must include variables in this attempt.

Let a_{1}, …, a_{m}, b_{1}, …, b_{r} ∈ ℝ′_{+}. We consider the equation

a_{1}^{x_{1}} ⋯ a_{m}^{x_{m}} = b_{1}^{y_{1}} ⋯ b_{r}^{y_{r}},  (9)

where x_{1}, …, x_{m} ∈ ℚ are variables and y_{1}, …, y_{r} ∈ ℚ \ {0} are constants. Factorizing b_{1}, …, b_{r} as in (1), we can reduce (9) to

a_{1}^{x_{1}} ⋯ a_{m}^{x_{m}} = q_{1}^{z_{1}} ⋯ q_{s}^{z_{s}},  (10)

where q_{1}, …, q_{s} ∈ ℙ are distinct and z_{1}, …, z_{s} ∈ ℚ \ {0} are constants. We define P and α_{ij} by (3) and (4), respectively; then (5) holds. We also denote

Q = {q_{1}, …, q_{s}}.

Theorem 8. If (10) has a solution, then

Q ⊆ P.  (11)

If (11) holds and

rank J_{a} = n,  (12)

where a is as in (6), then (10) has a solution.

Proof. First, assume that (10) has a solution. Let q_{i} ∈ Q. Since z_{i} ≠ 0, q_{i} divides the left-hand side of (10); so, q_{i} = p_{j} for some j, and (11) follows.

Second, assume that (11) and (12) hold. By reordering the indices of p_{1}, …, p_{n}, we can write Q = {p_{1}, …, p_{s}}. Let a and A_{a} be as in (6) and (7). Then

a_{i}^{x_{i}} = ∏_{j=1}^{n} p_{j}^{α_{ij}x_{i}}, i = 1, …, m,

and (10) reads

∏_{i=1}^{m} ∏_{j=1}^{n} p_{j}^{α_{ij}x_{i}} = ∏_{j=1}^{s} p_{j}^{z_{j}},

i.e.,

∏_{j=1}^{n} p_{j}^{∑_{i=1}^{m} α_{ij}x_{i}} = ∏_{j=1}^{s} p_{j}^{z_{j}}.  (13)

By Proposition 2, this is equivalent to

∑_{i=1}^{m} α_{ij}x_{i} = z_{j}, j = 1, …, s,

∑_{i=1}^{m} α_{ij}x_{i} = 0, j = s + 1, …, n.

In matrix form, this reads

A_{a}^{T} x = z,  (14)

where z_{s+1} = ⋯ = z_{n} = 0 and

x = ( x_{1} ; x_{2} ; ⋮ ; x_{m} ) , z = ( z_{1} ; z_{2} ; ⋮ ; z_{n} ) .

By Corollary 5 and (12),

rank A_{a}^{T} = rank A_{a} = rank J_{a} = n.  (15)

Therefore, A_{a}^{T} ℚ^{m} = ℚ^{n}, which implies that (14) has a solution. (Note that m ≥ n by (15).)

Corollary 9. Assume that m = n in (4). Then (10) has a unique solution if and only if (11) holds and det J_{a} ≠ 0.

Proof. The claim follows from (15).

4 Concluding remarks

We defined the arithmetic Jacobian matrix and multiplicative independence. We saw that the arithmetic Jacobian matrix relates to multiplicative independence in the same way as the ordinary Jacobian matrix relates to functional independence. We also noticed that the arithmetic Jacobian matrix and determinant play a role in establishing a certain kind of implicit function theorem somewhat similarly as the ordinary Jacobian matrix and determinant do in the ordinary implicit function theorem. For this purpose, we needed to introduce extra variables.

Another functional interpretation of arithmetic differentiation may be obtained as follows: If a ∈ ℝ′_{+} and p ∈ ℙ, then there are unique α ∈ ℚ and ã ∈ ℝ′_{+} such that a = ãp^{α} and p ∤ ã; in fact, α = ν_{p}(a). Further, a′_{p} = αãp^{α−1}. So, in studying arithmetic partial derivatives of first order, the primes behave, in a certain sense, like variables, and the rational numbers like functions.

Since the isomorphism ν brings the vector space structure to ℝ′_{+}, we can define all linear algebra concepts there. In particular, we have already studied the “arithmetic inner product”, see [2]. (In that paper, we considered only ℚ_{+} but every argument applies also for ℝ′_{+}.)

5 Acknowledgment

We thank the referee whose comments significantly improved our paper.

References

[1] E. J. Barbeau, Remarks on an arithmetic derivative, Canad. Math. Bull. 4 (1961), 117– 122.

[2] P. Haukkanen, M. Mattila, J. K. Merikoski, and T. Tossavainen, Perpendicularity in an Abelian group, Internat. J. Math. Math. Sci. 2013, Article ID 983607, 8 pp.

[3] P. Haukkanen, J. K. Merikoski, and T. Tossavainen, On arithmetic partial differential equations, J. Integer Sequences 19 (2016), Article 16.8.6.

[4] J. Kovič, The arithmetic derivative and antiderivative, J. Integer Sequences 15 (2012), Article 12.3.8.

[5] W. F. Newns, Functional dependence, Amer. Math. Monthly 74 (1967), 911–920.

[6] W. Y. Pong, Applications of differential algebra to algebraic independence of arithmetic functions, Acta Arith. 172 (2016), 149–173.

[7] W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976.

[8] V. Ufnarovski and B. Åhlander, How to differentiate a number, J. Integer Sequences 6 (2003), Article 03.3.4.

2010 Mathematics Subject Classification: Primary 11C20; Secondary 11A25, 15B36. Keywords: arithmetic derivative, arithmetic partial derivative, Jacobian matrix, Jacobian determinant, implicit function theorem, multiplicative independence.

(Concerned with sequences \underline{A000040} and \underline{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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