A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutions in non-negative integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(2n). We prove: (1) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite, (2) the conjecture implies that the question whether or not a Diophantine equation has only finitely many rational solutions is decidable with an oracle for deciding whether or not a Diophantine equation has a rational solution, (3) the conjecture implies that the question whether or not a Diophantine equation has only finitely many integer solutions is decidable with an oracle for deciding whether or not a Diophantine equation has an integer solution, (4) the conjecture implies that if a set M \subseteq N has a finite-fold Diophantine representation, then M is computable.Comment: 13 pages, section 7 expande

Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers?

Let B_n={x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let \xi(n) denote the smallest positive integer b such that for each system S \subseteq B_n with a unique solution in positive integers x_1,...,x_n, this solution belongs to [1,b]^n. Let g(1)=1, and let g(n+1)=2^{2^{g(n)}} for every positive integer n. We conjecture that \xi(n) \leq g(2n) for every positive integer n. We prove: (1) the function \xi: N\{0}-->N\{0} is computable in the limit; (2) if a function f:N\{0}-->N\{0} has a single-fold Diophantine representation, then there exists a positive integer m such that f(n)<\xi(n) for every integer n>m; (3) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation D(x_1,...,x_p)=0 and returns a positive integer d with the following property: for every positive integers a_1,...,a_p, if the tuple (a_1,...,a_p) solely solves the equation D(x_1,...,x_p)=0 in positive integers, then a_1,...,a_p \leq d; (4) the conjecture implies that if a set M \subseteq N has a single-fold Diophantine representation, then M is computable; (5) for every integer n>9, the inequality \xi(n)<(2^{2^{n-5}}-1)^{2^{n-5}}+1 implies that 2^{2^{n-5}}+1 is composite.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1309.268

A Brief History of Singlefold Diophantine Definitions

Consider an (m + 1)-ary relation R over the set N of natural numbers. Does there exist an arithmetical formula ZΘ(a0, . . . , am, x1, . . . , xK), not involving universal quantifiers, negation, or implication, such that the representation and univocity conditions, viz., (Formula Presented) are met by each tuple (Formula Presented). A priori, the answer may depend on the richness of the language of arithmetic: Even if solely addition and multiplication operators (along with the equality relator and with positive integer constants) are adopted as primitive symbols of the arithmetical signature, the graph R of any primitive recursive function is representable; but can representability be reconciled with univocity without calling into play one extra operator designating either the dyadic operation [b, n]↠ b n or just the monadic function n ↠ b n associated with a fixed integer b > 1? As a preparatory step toward a hoped-for positive answer to this question, one may consider replacing the exponentiation operator by a dyadic relator designating an exponential-growth relation (a notion made explicit by Julia Bowman Robinson in 1952). We will discuss the said univocity, aka ‘single-fold-ness’, issue-first raised by Yuri V. Matiyasevich in 1974-, framing it in historical context. © 2023 Copyright for this paper by its authors

The strong ABCABC conjecture over function fields (after McQuillan and Yamanoi)

The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the two independent proofs of the strong $abc$ conjecture over function fields given by McQuillan and Yamanoi. The first proof relies on tools from differential and algebraic geometry; the second relies on analytic and topological methods. They correspond respectively to the Nevanlinna and the Ahlfors approach to the Nevanlinna Second Main Theorem.Comment: 35 pages. This is the text of my Bourbaki talk in march 200