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Papers by Apoloniusz Tyszka

Scientific Annals of Computer Science, 2019

Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solutio... more Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smoryński's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. Let R be a subring of Q with or without 1. By H 10 (R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether or not the equation has a solution in R. We prove that a positive solution to H 10 (R) implies that the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. We show the converse implication for every infinite set R ⊆ Q such that there exist computable functions τ 1 , τ 2 : N → Z which satisfy (∀n ∈ N τ 2 (n) 0) ∧ τ 1 (n) τ 2 (n) : n ∈ N = R. This implication for R = N guarantees that Smoryński's theorem follows from Matiyasevich's theorem. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have a rational solution is not recursive. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have only finitely many rational solutions is not recursively enumerable. These conjectures are equivalent by our results for R = Q.

$Research paper thumbnail of A discrete form of the theorem that each field endomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="double-struck">R</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="double-struck">Q</mi><mtext>p</mtext></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbb{R}{\left( {\mathbb{Q}_{{\text{p}}} } \right)}</annotation></semantics></math>R(Qp) is the identity$

Aequationes mathematicae, 2006

Let R be a non-zero subring of Q (with or without 1) such that there exist computable functions τ... more Let R be a non-zero subring of Q (with or without 1) such that there exist computable functions τ 1 , τ 2 :

$Research paper thumbnail of A tuple (a_1,...,a_n) \in (N \{0})^n that for any n \geq 1290 satisfies max(a_1,...,a_n)>2^(2^(n-1)) and solves a system T \subseteq {x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} which has exactly one integer solution$

A tuple (a_1,...,a_n) \in (N \{0})^n that for any n \geq 1290 satisfies max(a_1,...,a_n)>2^(2^(n-1)) and solves a system T \subseteq {x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} which has exactly one integer solution

For any integer n \geq 1290, we define a system T \subseteq {x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_... more For any integer n \geq 1290, we define a system T \subseteq {x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} which has a unique integer solution (a_1,...,a_n). We prove that the numbers a_1,...,a_n are positive and
max(a_1,...,a_n)>2^(2^(n-1)).
The proof uses the fact that the number 21279−12^{1279}-121279−1 is prime.

$Research paper thumbnail of A tuple (a_1,...,a_n) \in (N\{0})^n that for any n \geq 2215 satisfies max(a_1,...,a_n)>2^(2^(n-1)) and solves a system T \subseteq {x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} which has exactly one integer solution$

A tuple (a_1,...,a_n) \in (N\{0})^n that for any n \geq 2215 satisfies max(a_1,...,a_n)>2^(2^(n-1)) and solves a system T \subseteq {x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} which has exactly one integer solution

For any integer n \geq 2215, we define a system T \subseteq {x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_... more For any integer n \geq 2215, we define a system T \subseteq {x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} which has a unique integer solution (a_1,...,a_n). We prove that the numbers a_1,...,a_n are positive and max(a_1,...,a_n)>2^(2^(n-1)). The proof uses the fact that the number 2^{2203}-1 is prime.

$Research paper thumbnail of A function f:N \setminus {0}->N \setminus {0} that cannot be bounded by a computable function and an infinite loop in MuPAD which takes as input a positive integer n and constantly returns f(n) after the first f(n) iterations$

"Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For a positive integer n,... more "Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let f(n) denote the greatest finite total number of solutions of a subsystem of E_n in integers x_1,...,x_n.
We prove: (1) the function f is strictly increasing, (2) if a non-decreasing function g from positive integers to positive integers satisfies f(n) \leq g(n) for any n, then a finite-fold Diophantine representation of g does not exist, (3) if the question of the title has a positive answer, then there is a computable strictly increasing function g from positive integers to positive integers such that f(n) \leq g(n) for any n and a finite-fold Diophantine representation of g does not exist."

$Research paper thumbnail of A function which does not have any finite-fold Diophantine representation and perhaps equals {(1,1)} \cup {(n,2^{2^{n-1}}): n \in {2,3,4,...}}$

Does there exist an algorithm which takes as input the length of a Diophantine equation and retur... more Does there exist an algorithm which takes as input the length of a Diophantine equation and returns a positive integer greater than the number of integer solutions, if these solutions form a finite set?

Drafts by Apoloniusz Tyszka

Let R be a non-zero subring of Q (with or without 1) such that there exist computable functions \... more Let R be a non-zero subring of Q (with or without 1) such that there exist computable functions \tau_1,\tau_2: N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). Matiyasevich's theorem states that there is no algorithm to decide whether or not a given Diophantine equation has a solution in non-negative integers. Smorynski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. We prove: (1) Smorynski's theorem easily follows from Matiyasevich's theorem, (2) Hilbert's Tenth Problem for solutions in R has a positive solution if and only if the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. ``Hilbert's Tenth Problem for solutions in R'' is the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether or not the equation has a solution with all unknowns taking values in R.

Let R be a non-zero subring of Q (with or without 1) such that there exist computable functions \... more Let R be a non-zero subring of Q (with or without 1) such that there exist computable functions \tau_1,\tau_2: N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). Matiyasevich's theorem states that there is no algorithm to decide whether or not a given Diophantine equation has a solution in non negative integers. Smorynski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. We prove: (1) Smorynski's theorem easily follows from Matiyasevich's theorem, (2) Hilbert's Tenth Problem for solutions in R has a positive solution if and only if the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. ``Hilbert's Tenth Problem for solutions in R'' is the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether or not the equation has a solution with all unknowns taking values in R.

Scientific Annals of Computer Science, 2019

Aequationes mathematicae, 2006

$Research paper thumbnail of A function which does not have any finite-fold Diophantine representation and perhaps equals {(1,1)} \cup {(n,2^{2^{n-1}}): n \in {2,3,4,...}}$

Let R be a non-zero subring of Q (with or without 1) such that there exist computable functions \... more Let R be a non-zero subring of Q (with or without 1) such that there exist computable functions \tau_1,\tau_2: N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). Matiyasevich's theorem states that there is no algorithm to decide whether or not a given Diophantine equation has a solution in non negative integers. Smorynski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. We prove: (1) Smorynski's theorem easily follows from Matiyasevich's theorem, (2) Hilbert's Tenth Problem for solutions in R has a positive solution if and only if the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. ``Hilbert's Tenth Problem for solutions in R'' is the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether or not the equation has a solution with all unknowns taking values in R.