Is there a computable upper bound on the heights of rational solutions of a Diophantine equation with a finite number of solutions?

Is there a computable upper bound on the heights of rational solutions of a Diophantine equation with a finite number of solutions?

Apoloniusz Tyszka University of Agriculture
Faculty of Production and Power Engineering
Balicka 116B, 30-149 Kraków, Poland
Email: rttyszka@cyf-kr.edu.pl
Abstract

The height of a rational number is denoted by and equals provided is written in lowest terms. The height of a rational tuple is denoted by and equals . Let . Let , and let for every positive integer . We conjecture: (1) if a system has only finitely many solutions in rationals , then each such solution satisfies ; (2) if a system has only finitely many solutions in non-negative rationals , then each such solution satisfies . We prove: (1) both conjectures imply that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of rational solutions, if the solution set is finite; (2) both conjectures imply that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution.

Diophantine equation which has only finitely many rational solutions, Hilbert’s Tenth Problem for , relative decidability, upper bound on the heights of rational solutions.

I Introduction

The height of a rational number is denoted by and equals provided is written in lowest terms. The height of a rational tuple is denoted by and equals . We attempt to formulate a conjecture which implies a positive answer to the following open problem: Is there an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of rational solutions, if the solution set is finite?

Ii Conjecture 1 and its equivalent form

Observation 1.

Only and solve the equation in integers (rationals, real numbers, complex numbers). For each integer , the following system

has exactly one integer (rational, real, complex) solution, namely .

Let

Conjecture 1.

If a system has only finitely many solutions in rationals , then each such solution satisfies

Observation 1 implies that the bound

cannot be decreased.

Conjecture 1 is equivalent to the following conjecture on rational arithmetic: if rational numbers satisfy

then there exist rational numbers such that

and for every

Theorem 1.

Conjecture 1 is true if and only if the execution of Flowchart 1 prints infinitely many numbers.

Flowchart 1: An infinite-time computation which

decides whether or not Conjecture 1 is true

Proof.

Let denote the set of all integers whose number of prime factors is divisible by . The claimed equivalence is true because the algorithm from Flowchart 1 applies a surjective function . ∎

Corollary 1.

Conjecture 1 can be written in the form , where is a computable predicate.

Iii Algebraic lemmas – part 1

Let denote the class of all rings, and let ng denote the class of all rings K that extend . Let

Lemma 1.

([15, p. 720]) Let . Assume that for each . We can compute a positive integer and a system which satisfies the following three conditions: Condition 1. If , then

Condition 2. If , then for each with , there exists a unique tuple such that the tuple solves . Condition 3. If denotes the maximum of the absolute values of the coefficients of , then

Conditions 1 and 2 imply that for each , the equation and the system have the same number of solutions in K.

Lemma 2.

([10, p. 100]) If and , then if and only if

Let , , and denote variables.

Lemma 3.

If and , then if and only if

(1)

and

(2)

We can express equations (1) and (2) as a system such that involves and new variables and consists of equations of the forms and .

Proof.

By Lemma 2, equation (1) is equivalent to

(3)

and equation (2) is equivalent to

(4)

The conjunction of equations (3) and (4) is equivalent to . The new variables express the following polynomials:

,     ,     ,     ,     ,     ,     ,

,     ,     ,     ,

,     ,     ,     ,

,   ,     ,

,     .

Lemma 4.

(cf. Observation 4) Let . Assume that for each . We can compute a positive integer and a system which satisfies the following two conditions: Condition 4. If , then

Condition 5. If , then for each with , there exists a unique tuple such that the tuple solves . Conditions 4 and 5 imply that for each , the equation and the system have the same number of solutions in K.

Proof.

Let the system be given by Lemma 1. For every ,

Therefore, if there exists such that the equation belongs to , then we introduce a new variable and replace in each equation of the form by the equations , , . Next, we apply Lemma 3 to each equation of the form that belongs to and replace in each such equation by an equivalent system of equations of the forms and . ∎

Iv The main consequence of Conjecture 1

Theorem 2.

Conjecture 1 implies that there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of rational solutions, if the solution set is finite.

Proof.

It follows from Lemma 4 for . The claim of Theorem 2 also follows from Observation 4. ∎

Corollary 2.

Conjecture 1 implies that the set of all Diophantine equations which have infinitely many rational solutions is recursively enumerable. Assuming Conjecture 1, a single query to the halting oracle decides whether or not a given Diophantine equation has infinitely many rational solutions. By the Davis-Putnam-Robinson-Matiyasevich theorem, the same is true for an oracle that decides whether or not a given Diophantine equation has an integer solution.

For many Diophantine equations we know that the number of rational solutions is finite by Faltings’ theorem. Faltings’ theorem tells that certain curves have finitely many rational points, but no known proof gives any bound on the sizes of the numerators and denominators of the coordinates of those points, see [6, p. 722]. In all such cases Conjecture 1 allows us to compute such a bound. If this bound is small enough, that allows us to find all rational solutions by an exhaustive search. For example, the equation has only finitely many rational solutions ([9, p. 212]). The known rational solutions are: , , , , , , , , , , , , , , , , and the existence of other solutions is an open question, see [12, pp. 223–224]. The system

is equivalent to . By Conjecture 1, . Therefore, . Assuming that Conjecture 1 holds, the following MuPAD code finds all rational solutions of the equation .


solutions:=:for i from -256 to 256 dofor j from 1 to 256 dox:=i/j:y:=4*x^5-4*x+1:p:=numer(y):q:=denom(y):if numlib::issqr(p) and numlib::issqr(q) thenz1:=sqrt(p/q):z2:=-sqrt(p/q):y1:=(z1+1)/2:y2:=(z2+1)/2:solutions:=solutions union [x,y1],[x,y2]:end_if:end_for:end_for:print(solutions):\@endparenv

The code solves the equivalent equation

and displays the already presented solutions.

MuPAD is a general-purpose computer algebra system. The commercial version of MuPAD is no longer available as a stand-alone product, but only as the Symbolic Math Toolbox of MATLAB. Fortunately, this code can be executed by MuPAD Light, which was offered for free for research and education until autumn 2005.

V Algebraic lemmas – part 2

Lemma 5.

Lemmas 2 and 3 are not necessary for proving that in the rational domain each Diophantine equation is equivalent to a system of equations of the forms and .

Proof.

By Lemma 1, an arbitrary Diophantine equation is equivalent to a system , where and can be computed. If there exists such that the equation belongs to , then we introduce a new variable and replace in each equation of the form by the equations , , and . For each rational number , we have and . Hence, for each rational numbers , , ,

We transform the last equation into an equivalent system in such a way that the variables correspond to the following rational expressions:

In this way, we replace in each equation of the form by an equivalent system of equations of the forms and . ∎

The next lemma enable us to prove Theorem 2 without using Lemma 4.

Lemma 6.

For solutions in a field, each system is equivalent to , where each is a system of equations of the forms and .

Proof.

Acting as in the proof of Lemma 5, we eliminate from  all equations of the form . Let denote the number of equations of the form that belong to . We can assume that . Let the variables , , , , , and be new. Let

and let

The system expresses that and . The system expresses that and . Therefore, . We have described a procedure which transforms into and . We iterate this procedure for and and finally obtain the systems without equations of the form . The systems satisfy and they contain only equations of the forms and . ∎

Vi Systems which have infinitely many rational solutions

Lemma 7.

([11, p. 391]) If has an odd exponent in the prime factorization of a positive integer , then can be written as the sum of three squares of integers.

Lemma 8.

For each positive rational number , or can be written as the sum of three squares of rational numbers.

Proof.

We find positive integers and with . If has an odd exponent in the prime factorization of , then by Lemma 7 there exist integers , , such that . Hence,

If has an even exponent in the prime factorization of , then by Lemma 7 there exist integers , , such that . Hence,

Lemma 9.

A rational number can be written as the sum of three squares of rational numbers if and only if there exist rational numbers , , such that .

Proof.

Let . Of course,

We prove that for each rational numbers , , there exist rational numbers , , such that . Without loss of generality we can assume that . If , then and . If , then and . ∎

Lemma 10.

([1, p. 125]) The equation has infinitely many solutions in positive rationals and each such solution satisfies .

Theorem 3.

There exists a system such that has infinitely many solutions in rationals and each such solution has height greater than .

Proof.

We define:

Let denote the set of all positive rationals such that the system

is solvable in rationals. Let denote the set of all positive rationals such that the system

is solvable in rationals. Lemma 10 implies that the set is infinite. By Lemma 8, . Therefore, is infinite (Case 1) or is infinite (Case 2).

Case 1. In this case the system

has infinitely many rational solutions. By this and Lemma 9, the system

has infinitely many rational solutions. We transform the above system into an equivalent system in such a way that the variables correspond to the following rational expressions:

,  ,  ,  ,  ,  ,  ,  ,

,  ,  ,  ,  ,  ,  ,  ,  ,  ,

,  ,  ,  ,  ,  ,  ,  ,  .

The system has infinitely many solutions in rationals . Lemma 10 implies that each rational tuple that solves satisfies

Since , and the proof for Case 1 is complete.

Case 2. In this case the system

has infinitely many rational solutions. By this and Lemma 9, the system

has infinitely many rational solutions. We transform the above system into an equivalent system in such a way that the variables correspond to the following rational expressions:

,

,

.

The system has infinitely many solutions in rationals . Lemma 10 implies that each rational tuple that solves satisfies

For a positive integer , let denote the smallest positive integer such that each system solvable in rationals has a rational solution whose height is not greater than . Obviously, . Observation 1 implies that for every integer . Theorem 3 implies that .

Theorem 4.

The function is computable in the limit.

Proof.

Let us agree that the empty tuple has height . For a positive integer and a tuple

let denote the successor of in the co-lexicographic order on . Flowchart 2 illustrates an infinite-time computation of .

Flowchart 2: An infinite-time computation of

Vii Conjecture 2 and its equivalent form

Let denote the integer part function.

Lemma 11.

For every non-negative real numbers and , implies that .

Proof.

For every non-negative real numbers and , implies that . ∎

Let , and let for every positive integer . Let , and let for every positive integer .

Conjecture 2.

If a system has only finitely many solutions in non-negative rationals , then each such solution satisfies .

Observations 2 and 3 justify Conjecture 2.

Observation 2.

For every system which involves all the variables , the following new system

is equivalent to . If the system has only finitely many solutions in non-negative rationals , then the new system has only finitely many solutions in non-negative rationals .

Proof.

It follows from Lemma 11. ∎

Observation 3.

For every positive integer , the following system

has exactly two solutions in non-negative rationals, namely and . The second solution has greater height.

Conjecture 2 is equivalent to the following conjecture on the arithmetic of non-negative rationals: if non-negative rational numbers satisfy , then there exist non-negative rational numbers such that

and for every

Theorem 5.

Conjecture 2 is true if and only if the execution of Flowchart 3 prints infinitely many numbers.

Flowchart 3: An infinite-time computation which

decides whether or not Conjecture 2 is true

Proof.

Let denote the set of all integers whose number of prime factors is divisible by . The claimed equivalence is true because the algorithm from Flowchart 3 applies a surjective function from to . ∎

Corollary 3.

Conjecture 2 can be written in the form , where is a computable predicate.

Viii Algebraic lemmas – part 3

Lemma 12.

(cf. [10, p. 100]) For every non-negative real numbers , if and only if

(5)
Proof.

The left side of equation (5) minus the right side of equation (5) equals . ∎

Lemma 13.

In non-negative rationals, the equation is equivalent to a system which consists of equations of the forms and .

Proof.

It follows from Lemma 12. ∎

Lemma 14.

Let . Assume that for each . We can compute a positive integer and a system which satisfies the following two conditions: Condition 6. For every non-negative rationals ,

Condition 7. If non-negative rationals satisfy , then there exists a unique tuple such that the tuple solves . Conditions 6 and 7 imply that the equation and the system have the same number of solutions in non-negative rationals.

Proof.

We write down the polynomial and replace each coefficient by the successor of its absolute value. Let denote the obtained polynomial. The polynomials and have positive integer coefficients. The equation is equivalent to

There exist a positive integer and a finite non-empty list such that