A logical approach to the problem of
representation of integers by systems of
diagonal forms
Javier Utreras
University of Manchester
December 17th, 2010
Javier Utreras
Representation of integers by systems of diagonal forms
Hilbert’s Tenth Problem...
Problem (Hilbert’s Tenth Problem)
Find an algorithm that decides whether a Diophantine equation
has an integral solution or not.
Javier Utreras
Representation of integers by systems of diagonal forms
Hilbert’s Tenth Problem...
Problem (Hilbert’s Tenth Problem)
Find an algorithm that decides whether a Diophantine equation
has an integral solution or not.
Theorem (Davis, Putnam, Robinson, Matiyasevic)
Such an algorithm does not exist.
Javier Utreras
Representation of integers by systems of diagonal forms
... and some related results
Facts
Javier Utreras
Representation of integers by systems of diagonal forms
... and some related results
Facts
There exists an algorithm that decides whether a system of
Diophantine linear equations has an integral solution or not.
Javier Utreras
Representation of integers by systems of diagonal forms
... and some related results
Facts
There exists an algorithm that decides whether a system of
Diophantine linear equations has an integral solution or not.
There exists an algorithm that decides whether a Diophantine
quadratic equation has an integral solution or not.
Javier Utreras
Representation of integers by systems of diagonal forms
... and some related results
Facts
There exists an algorithm that decides whether a system of
Diophantine linear equations has an integral solution or not.
There exists an algorithm that decides whether a Diophantine
quadratic equation has an integral solution or not.
There exists no algorithm that decides whether a system of
Diophantine quadratic equations has an integral solution or
not.
Javier Utreras
Representation of integers by systems of diagonal forms
The open problem in between
Problem (Representation of integers by systems of diagonal
quadratic forms)
Determine whether there exists an algorithm that decides the
“existence of integer solutions problem” for systems of the form
m
X
aij xi2 = bj ,
b = 1, . . . n
i=1
(with aij , bj ∈ Z).
Javier Utreras
Representation of integers by systems of diagonal forms
The open problem in between
Consider the first-order language
L2 = {0, 1, +, P2 }.
Interpret P2 over Z as the predicate “is a square”.
Javier Utreras
Representation of integers by systems of diagonal forms
The open problem in between
Consider the first-order language
L2 = {0, 1, +, P2 }.
Interpret P2 over Z as the predicate “is a square”.
Problem (Representation of integers by systems of diagonal
quadratic forms)
Determine whether the positive-existential theory of Z in the
language L2 is decidable.
Javier Utreras
Representation of integers by systems of diagonal forms
Extending the L2 language
Javier Utreras
Representation of integers by systems of diagonal forms
Extending the L2 language
Theorem (U.)
Let L be a first order language containing L2 . If there exists a
positive existential L-formula φ(x, y ) that satisfies
(i) if y = x 2 , then N satisfies φ(x, y ), and
(ii) if N models φ(x, y ) and x is nonzero, then x divides y ,
then the positive existential theory of N in the language L is
undecidable.
Javier Utreras
Representation of integers by systems of diagonal forms
Extending the L2 language
Idea of proof.
We need to define the function x 7→ x 2 over N in L.
Javier Utreras
Representation of integers by systems of diagonal forms
Extending the L2 language
Idea of proof.
We need to define the function x 7→ x 2 over N in L.
Given x ∈ N, we look for a natural number y satisfying
N |= P2 (y );
Javier Utreras
Representation of integers by systems of diagonal forms
Extending the L2 language
Idea of proof.
We need to define the function x 7→ x 2 over N in L.
Given x ∈ N, we look for a natural number y satisfying
N |= P2 (y );
N |= P2 (y + 2x + 1);
Javier Utreras
Representation of integers by systems of diagonal forms
Extending the L2 language
Idea of proof.
We need to define the function x 7→ x 2 over N in L.
Given x ∈ N, we look for a natural number y satisfying
N |= P2 (y );
N |= P2 (y + 2x + 1);
N |= φ(x, y ) ∧ φ(x + 1, y + 2x + 1).
Javier Utreras
Representation of integers by systems of diagonal forms
Extending the L2 language
Idea of proof.
We need to define the function x 7→ x 2 over N in L.
Given x ∈ N, we look for a natural number y satisfying
N |= P2 (y );
N |= P2 (y + 2x + 1);
N |= φ(x, y ) ∧ φ(x + 1, y + 2x + 1).
It is not difficult to show then that said y is unique.
Javier Utreras
Representation of integers by systems of diagonal forms
Applications
Javier Utreras
Representation of integers by systems of diagonal forms
Applications
Example
Consider the language {0, 1, +, P2 , |}, where | is interpreted over Z
as the divisibility relation. As the formula
x|y
satisfies the hypotheses of the theorem, the positive existential
theory of Z in this language is undecidable.
Javier Utreras
Representation of integers by systems of diagonal forms
Applications
Example
Consider the language {0, 1, +, P2 , |}, where | is interpreted over Z
as the divisibility relation. As the formula
x|y
satisfies the hypotheses of the theorem, the positive existential
theory of Z in this language is undecidable.
Remark
On the other hand, the positive existential of Z in the language
{0, 1, +, |} is decidable.
Javier Utreras
Representation of integers by systems of diagonal forms
Applications
Example
For k ≥ 1, consider the languages L2k = {0, 1, +, P2 , Rk }, where
each Rk is interpreted over N as the set of pairs (x, y ) such that
there exists an integer t satisfying
y = tx, and
every prime number p that divides t is not greater that
√
k
x.
The positive existential theories of N in each of these languages L2k
are all undecidable.
Javier Utreras
Representation of integers by systems of diagonal forms
Applications
Example
For k ≥ 1, consider the languages L2k = {0, 1, +, P2 , Rk }, where
each Rk is interpreted over N as the set of pairs (x, y ) such that
there exists an integer t satisfying
y = tx, and
every prime number p that divides t is not greater that
√
k
x.
The positive existential theories of N in each of these languages L2k
are all undecidable.
Remark
For each L2k a different formula satisfying the hypotheses of the
theorem is found.
Javier Utreras
Representation of integers by systems of diagonal forms
Higher powers
Problem (Representation of integers by systems of diagonal
k-th power forms)
For each k ≥ 2, determine whether there exists an algorithm that
decides the “existence of integer solutions problem” for systems of
the form
m
X
aij xik = bj , b = 1, . . . n
i=1
(with aij , bj ∈ Z).
Javier Utreras
Representation of integers by systems of diagonal forms
Higher powers
Problem (Representation of integers by systems of diagonal
k-th power forms)
For each k ≥ 2, determine whether there exists an algorithm that
decides the “existence of integer solutions problem” for systems of
the form
m
X
aij xik = bj , b = 1, . . . n
i=1
(with aij , bj ∈ Z).
Remark
As before, for each k ≥ 2 this is equivalent to the decidability
problem for the positive existential theory of Z in the language
Lk = {0, 1, +, Pk }, where Pk is interpreted as the predicate “is a
k-th power”.
Javier Utreras
Representation of integers by systems of diagonal forms
Higher powers
Theorem (U.)
The positive existential theories of N in the languages
{0, 1, +, Pk , |} are all undecidable.
Javier Utreras
Representation of integers by systems of diagonal forms
Higher powers
Idea of proof.
Fix k ≥ 2. We want to define a formula φ(x, y ) such that, given
x ∈ N, if N models φ(x, y ) then
N |= Pk (y );
Javier Utreras
Representation of integers by systems of diagonal forms
Higher powers
Idea of proof.
Fix k ≥ 2. We want to define a formula φ(x, y ) such that, given
x ∈ N, if N models φ(x, y ) then
N |= Pk (y );
N |= ∃z(z ≤ t(x) ∧ Pk (y + z)) for some definable t(x);
Javier Utreras
Representation of integers by systems of diagonal forms
Higher powers
Idea of proof.
Fix k ≥ 2. We want to define a formula φ(x, y ) such that, given
x ∈ N, if N models φ(x, y ) then
N |= Pk (y );
N |= ∃z(z ≤ t(x) ∧ Pk (y + z)) for some definable t(x);
on the other hand, we do not want max{y : N |= φ(x, y )}
growing “too fast” with respect to x; here the function t(x)
comes into play. If it is suitably defined we manage to bound
k2
k
the desired maximum by 2 k−1 x k−1 .
Javier Utreras
Representation of integers by systems of diagonal forms
Higher powers
Idea of proof.
Fix k ≥ 2. We want to define a formula φ(x, y ) such that, given
x ∈ N, if N models φ(x, y ) then
N |= Pk (y );
N |= ∃z(z ≤ t(x) ∧ Pk (y + z)) for some definable t(x);
on the other hand, we do not want max{y : N |= φ(x, y )}
growing “too fast” with respect to x; here the function t(x)
comes into play. If it is suitably defined we manage to bound
k2
k
the desired maximum by 2 k−1 x k−1 .
Finally, using a theorem of Kosovskii we define the function
x 7→ x 2 by iterating the previous formula and adding some extra
divisibility conditions.
Javier Utreras
Representation of integers by systems of diagonal forms
References
N. K. Kosovskii,
On solutions of systems consisting of both word equations and
word length inequalities.
Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.
(POMI) 40 (1974), 24–29 (in Russian);
English transl.: J. Math. Sci. (N. Y.) 8 (1977), no. 3, 262–265.
J. Utreras,
A logical approach to the problem of representation of integers
by systems of diagonal forms.
to appear in the Bulletin of the London Mathematical Society.
Javier Utreras
Representation of integers by systems of diagonal forms