Diophantine set

Diophantine set

A set

S

is said to be

Diophantine

if

•

it is a subset of

ℕ

n

, the set of all

n

-tuples of

positive integers

, and

•

there is a

polynomial

p

over

ℤ

in

n

+

k

variables,

k

≥

0

, such that

x

∈

S

iff there is some

y

∈

ℕ

k

, such that

p

⁢

(

x

,

y

)

=

0

.

So

S

can be thought of as a set such that, there is a

Diophantine equation

p

=

0

and a non-negative

integer

k

, so that when each

element

in

S

is “combined” with some

k

-tuple, makes up a solution to a Diophantine equation

p

=

0

. In other words, if

f

n

n

+

k

:

ℕ

n

+

k

→

ℕ

n

is a projection function given by

f

n

n

+

k

⁢

(

x

,

y

)

=

x

where

x

∈

ℕ

n

and

y

∈

ℕ

k

, then

S

is a

Diophantine set

iff

S

=

f

n

n

+

k

⁢

(

Z

)

, where

Z

is the

zero set

of some Diophantine equation

p

=

0

. Equivalently, a set

S

∈

ℕ

n

is Diophantine if there is a

p

∈

ℤ

⁢

[

X

1

,

…

,

X

n

+

k

]

, such that

S

=

{

x

∈

ℕ

n

∣

∃

y

∈

ℕ

k

⁢

p

⁢

(

x

,

y

)

=

0

}

.

For example,

ℕ

itself is Diophantine, for the polynomial

p

⁢

(

x

,

y

)

=

x

-

y

works. Another

trivial

example: the

set of all positive integers

divisible

by

3

is Diophantine, for the polynomial

p

⁢

(

x

,

y

)

=

x

-

3

⁢

y

works.

For a less trivial example, let us show that the set of all triples

(

a

,

b

,

c

)

such that

a

≤

b

≤

c

is Diophantine. For the

inequality

a

≤

b

, let

p

⁢

(

x

1

,

x

2

,

y

)

=

x

2

-

x

1

-

(

y

-

1

)

. Then the

sentence

∃

y

⁢

p

⁢

(

x

1

,

x

2

,

y

)

=

0

is

equivalent

to

x

1

≤

x

2

. Similarly, for the inequality

b

≤

c

, we have the same polynomial

p

. Putting the two inequality together amounts to setting

q

⁢

(

x

1

,

x

2

,

x

3

,

y

1

,

y

2

)

=

p

⁢

(

x

1

,

x

2

,

y

1

)

2

+

p

⁢

(

x

2

,

x

3

,

y

2

)

2

. Thus, the sentence

∃

(

y

1

,

y

2

)

⁢

q

⁢

(

x

,

y

)

=

0

, where

x

=

(

x

1

,

x

2

,

x

3

)

and

y

=

(

y

1

,

y

2

)

is the same as the inequality

x

1

≤

x

2

≤

x

3

.

Some other Diophantine sets are:

•

the set

A

=

B

∪

C

, where

B

and

C

are Diophantine

•

the set

A

=

B

∩

C

, where

B

and

C

are Diophantine

•

the set

{

a

∣

a

≡

b

(

mod

n

)

}

•

the set of

composite numbers

•

the set of

prime numbers

•

the set of powers of a positive integer

{

m

n

∣

n

=

0

,

1

,

…

}

Remark

. Associated with the

concept

of a Diophantine set is that of a

Diophantine function

: a

function

f

is said to be

Diophantine

if its graph

{

(

x

,

f

⁢

(

x

)

)

∣

x

∈

dom

⁡

(

f

)

}

is a Diophantine set. Some well-know Diophantine functions are the

exponential functions

f

⁢

(

x

)

=

n

x

and the

factorial function

f

⁢

(

x

)

=

x

!

, where

n

,

x

are positive integers.

It turns out that a function is Diophantine iff it is

recursive

. From this, it is possible to prove that Hilbert’s 10th problem is unsolvable.

The idea behind using Diophantine sets to prove the unsolvability of Hilbert’s 10th problem comes from Yuri Matiyaseviĉ, and hence the

theorem

is known as Matiyaseviĉ’s theorem.

References

1

M Davis,

Computability and Unsolvability

. Dover Publications, Inc. New York, 1982

Title

Diophantine set

Canonical name

DiophantineSet

Date of creation

2013-03-22 18:02:50

Last modified on

2013-03-22 18:02:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D99

Classification

msc 03D80

Defines

Diophantine function

Generated on Thu Feb 8 20:06:43 2018 by

LaTeXML