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Matiyasevich's theorem

Matiyasevich's theorem, proven in 1970 by Yuri Matiyasevich, implies that Hilbert's tenth problem is unsolvable. This problem is the challenge to find a general algorithm which can decide whether a given system of Diophantine equations (polynomials with integer coefficients) has a solution among the integers. David Hilbert posed the problem in his 1900 address to the International Congress of Mathematicians.

A typical system of diophantine equations looks like this:

3x2y − 7y2z3 = 18
− 7y2 + 8z2 = 0
The question is whether there exist integers x, y and z which satisfy both equations simultaneously. It turns out that it is always equivalent to ask whether a single Diophantine equation with several variables has any solutions among the natural numbers. For instance, the above system is solvable over the integers if and only if the following equation is solvable over the natural numbers:
( 3(x1x2)2(y1y2) − 7(y1y2)2(z1z2)3 − 18 )2 + ( −7(y1y2)2 + 8(z1z2)2)2 = 0.

Matiyasevich utilized an ingenious trick involving Fibonacci numbers in order to show that solutions to Diophantine equations may grow exponentially. Earlier work by Julia Robinson, Martin Davis and Hilary Putnam had shown that this suffices to show that no general algorithm deciding the solvability of Diophantine equations can exist.

Later work has shown that the question of solvability of a Diophantine equation is undecidable even if the equation only has 9 natural number variables (Matiyasevich, 1977) or 11 integer variables (Zhi Wei Sun, 1992).

Matiyasevich's theorem itself is somewhat more general than the unsolvability of the Tenth Problem. It says:

Every recursively enumerable set is Diophantine.

A set S of integers is recursively enumerable precisely if there is an algorithm that behaves as follows: When given as input an integer n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n,x1,...,xk) such that an integer n is in S if and only if there exist some integers x1,...,xk such that f(n,x1,...,xk)=0. The conjunction of Matiyasevich's theorem with a result discovered in the 1930s implies that a solution to Hilbert's tenth problem is impossible. The result discovered in the 1930s by several logicians can be stated by saying that some recursively enumerable sets are non-recursive. In this context, a set S of integers is called "recursive" if there is an algorithm that, when given as input an integer n, returns as output a correct yes-or-no answer to the question of whether n is a member of S.

(It is amusing to observe that one of the very few places in modern mathematics where an argument that takes exactly the form of an old-fashioned Aristotelian syllogism is of great interest and is not contemptuously dismissed as uninteresting because trivial. That argument is as follows.

(Major premise): All recursively enumerable sets are Diophantine.
(Minor premise): Some recursively enumerable sets are non-recursive.
(Conclusion): Therefore some Diophantine sets are non-recursive.

The conclusion of this syllogism is what entails that Hilbert's 10th problem cannot be solved.)

Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are unsolvable.

One can also derive the following stronger form of Gödel's incompleteness theorem from Matiyasevich's result:

Corresponding to any given axiomatization of number theory, one can explicitly construct a Diophantine equation which has no solutions, but such that this fact cannot be proved within the given axiomatization.