Detecting algebraic (in)dependence of explicitly presented functions (some applications of Nevanlinna theory to mathematical logic)
Authors: Gurevič R. H. 
Published: June 2012

We consider algebraic relations between explicitly presented analytic functions with particular emphasis on Tarski’s high school algebra problem. The part not related directly to Tarski’s high school algebra problem. Let

U
U

be a connected complex-analytic manifold. Denote by

F

(
U
)

\mathcal {F}(U)

the minimal field containing all functions meromorphic on

U
U

and closed under exponentiation

f
↦

e
f

f \mapsto {e^f}

. Let

f
j

∈

F

(
U
)

{f_j} \in \mathcal {F}(U)

,

p
j

∈

M

(
U
)
−
{
0
}

{p_j} \in \mathcal {M}(U) - \{ 0\}

for

1
≤
j
≤
m

1 \leq j \leq m

, and

g
k

∈

F

(
U
)

{g_k} \in \mathcal {F}(U)

,

q
k

∈

M

(
U
)
−
{
0
}

{q_k} \in \mathcal {M}(U) - \{ 0\}

for

1
≤
k
≤
n

1 \leq k \leq n

(where

M

(
U
)

\mathcal {M}(U)

is the field of functions meromorphic on

U
U

). Let

f
i

−

f
j

∉

H

(
U
)

{f_i} - {f_j} \notin \mathcal {H}(U)

for

i
≠
j

i \ne j

and

g
k

−

g
l

∉

H

(
U
)

{g_k} - {g_l} \notin \mathcal {H}(U)

for

k
≠
1

k \ne 1

(where

H

(
U
)

\mathcal {H}(U)

is the ring of functions holomorphic on

U
U

). If all zeros and singularities of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h equals StartFraction sigma-summation Underscript j equals 1 Overscript m Endscripts p Subscript j Baseline e Superscript f Super Subscript j Superscript Baseline Over sigma-summation Underscript k equals 1 Overscript n Endscripts q Subscript k Baseline e Superscript g Super Subscript k Superscript Baseline EndFraction">

h
=

∑

j
=
1

m

p
j

e

f
j

∑

k
=
1

n

q
k

e

g
k

h = \frac {{\sum \nolimits _{j = 1}^m {{p_j}{e^{{f_j}}}} }} {{\sum \nolimits _{k = 1}^n {{q_k}{e^{{g_k}}}} }}

\]
are contained in an analytic subset of

U
U

then

m
=
n

m = n

and there exists a permutation

σ
\sigma

of

{
1
,
…
,
m
}

\{ 1, \ldots ,m\}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h equals left-parenthesis p Subscript j Baseline slash q Subscript sigma left-parenthesis j right-parenthesis Baseline right-parenthesis dot e Superscript f Super Subscript j Superscript minus g Super Subscript sigma left-parenthesis j right-parenthesis">

h
=
(

p
j

/

q

σ
(
j
)

)
⋅

e

f
j

−

g

σ
(
j
)

h = ({p_j}/{q_{\sigma (j)}}) \cdot {e^{{f_j} - {g_{\sigma (j)}}}}

for

1
≤
j
≤
m

1 \leq j \leq m

. When

h
∈

M

(
U
)

h \in \mathcal {M}(U)

, additionally

f
j

−

g

σ
(
j
)

∈

H

(
U
)

{f_j} - {g_{\sigma (j)}} \in \mathcal {H}(U)

for all

j
j

. On Tarski’s high school algebra problem. Consider

L
=
{

L = \{

-terms in variables and

1
1

,

+
+

,

⋅
\cdot

,

↑
}

\uparrow \}

, where

↑:
a

\uparrow :a

,

b
↦

a
b

b \mapsto {a^b}

for positive

a
a

,

b
b

. Each term

t
∈
L

t \in L

naturally determines a function

t
¯

\bar t

:

(

R

+

)
n

→

R

−

{({{\mathbf {R}}_ + })^n} \to {{\mathbf {R}}_ - }

, where

n
n

is the number of variables involved. For

S
⊂
L

S \subset L

put

S
¯

=
{

t
¯

|

t
∈
S
}

\bar S = \{ \bar t|t \in S\}

. (i) We describe the algebraic structure of

Λ
¯

\bar \Lambda

and

L

¯

\bar {\mathcal {L}}

, where

Λ
=
{
t
∈
L

|

\Lambda = \{ t \in L|

if

u
↑
v

u \uparrow v

occurs as a subterm of

t
t

then either

u
u

is a variable or

u
u

contains no variables at all} , and

L

=
{
t
∈
L

|

\mathcal {L} = \{ t \in L|

if

u
↑
v

u \uparrow v

occurs as a subterm of

t
t

then

u
∈
Λ
}

u \in \Lambda \}

. Of these,

Λ
¯

\bar \Lambda

is a free semiring with respect to addition and multiplication but

L

¯

\bar {\mathcal {L}}

is free only as a semigroup with respect to addition. A function

t
¯

∈

S
¯

\bar t \in \bar S

is called

+
+

-prime in

S
¯

\bar S

if

t
¯

≠

u
¯

+

v
¯

\bar t \ne \bar u + \bar v

for all

u
u

,

v
∈
S

v \in S

and is called multiplicatively prime in

S
¯

\bar S

if

t
¯

=

u
¯

⋅

v
¯

⇒

u
¯

=
1

\bar t = \bar u \cdot \bar v \Rightarrow \bar u = 1

or

v
¯

=
1

\bar v = 1

for

u
u

,

v
∈
S

v \in S

. A function is called

(
+
,
⋅
)

( + , \cdot )

-prime in

S
¯

\bar S

if it is both

+
+

-prime and multiplicatively prime in

S
¯

\bar S

. A function in

Λ
¯

\bar \Lambda

is said to have content

1
1

if it is not divisible by constants in

N

−
{
1
}

{\mathbf {N}} - \{ 1\}

or by

≠
1
 
(
+
,
⋅
)

\ne 1\ ( + , \cdot )

-primes of

Λ
¯

\bar \Lambda

. The product of functions of content

1
1

has content

1
1

. Let

P
P

be the multiplicative subsemigroup of

Λ
¯

\bar \Lambda

of functions of content

1
1

. Then

L

¯

\bar {\mathcal {L}}

as a semiring is isomorphic to the semigroup semiring

Λ
¯

(

⊕
f

P
f

)

\bar \Lambda ({ \oplus _f}{P_f})

, where each

P
f

{P_f}

is a copy of

P
P

and

f
f

ranges over the

≠
1

+

\ne 1\; +

-primes of

L

¯

\bar {\mathcal {L}}

. (ii) We prove that if

t
t

,

u
∈

L

u \in \mathcal {L}

and

R

+

⊢
t
=
u

{{\mathbf {R}}_ + } \vdash t = u

(i.e., if

t
¯

=

u
¯

\bar t = \bar u

) then Tarski’s "high school algebra" identities

⊢
t
=
u

\vdash t = u

. This result covers a conjecture of C. W. Henson and L. A. Rubel. (Note: this result does not generalize to arbitrary

t
t

,

u
∈
L

u \in L

. Moreover, the equational theory of

(

R

+

;

1
,
+
,
⋅
,
↑
)

({{\mathbf {R}}_ + };\;1, + , \cdot , \uparrow )

is not finitely axiomatizable.