Диофант теңдеулерінің шешілмеуі туралы

On solvability of Diophantine equations
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values. For example, the Diophantine equation has an integer solution: By contrast, the Diophantine equation has no such solution. Hilbert's tenth problem has been solved, and it has a negative answer: such a general algorithm cannot exist. This is the result of combined work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson that spans 21 years, with Matiyasevich completing the theorem in 1970. The theorem is now known as Matiyasevich's theorem or the MRDP theorem (an initialism for the surnames of the four principal contributors to its solution). When all coefficients and variables are restricted to be positive integers, the related problem of polynomial identity testing becomes a decidable (exponentiation free) variation of Tarski's high school algebra problem, sometimes denoted

We may speak of the degree of a Diophantine set as being the least degree of a polynomial in an equation defining that set. Similarly, we can call the dimension of such a set the fewest unknowns in a defining equation. Because of the existence of a universal Diophantine equation, it is clear that there are absolute upper bounds to both of these quantities, and there has been much interest in determining these bounds. Already in the 1920s Thoralf Skolem showed that any Diophantine equation is equivalent to one of degree 4 or less. His trick was to introduce new unknowns by equations setting them equal to the square of an unknown or the product of two unknowns. Repetition of this process results in a system of second degree equations; then an equation of degree 4 is obtained by summing the squares. So every Diophantine set is trivially of degree 4 or less. It is not known whether this result is best possible. Julia Robinson and Yuri Matiyasevich showed that every Diophantine set has dimension no greater than 13. Later, Matiyasevich sharpened their methods to show that 9 unknowns suffice. Although it may well be that this result is not the best possible, there has been no further progress. So, in particular, there is no algorithm for testing Diophantine equations with 9 or fewer unknowns for solvability in natural numbers. For the case of rational integer solutions (as Hilbert had originally posed it), the 4 squares trick shows that there is no algorithm for equations with no more than 36 unknowns. But Zhi Wei Sun showed that the problem for integers is unsolvable even for equations with no more than 11 unknowns. Martin Davis studied algorithmic questions involving the number of solutions of a Diophantine equation. Hilbert's tenth problem asks whether or not that number is 0. Let and let be a proper non empty subset of Davis proved that there is no algorithm to test a given Diophantine equation to determine whether the number of its solutions is a member of the set Thus there is no algorithm to determine whether the number of solutions of a Diophantine equation is finite, odd, a perfect square, a prime, etc. The proof of the MRDP theorem has been formalized in Coq.

Although Hilbert posed the problem for the rational integers, it can be just as well asked for many rings (in particular, for any ring whose number of elements is countable). Obvious examples are the rings of integers of algebraic number fields as well as the rational numbers. There has been much work on Hilbert's tenth problem for the rings of integers of algebraic number fields. Basing themselves on earlier work by Jan Denef and Leonard Lipschitz and using class field theory, Harold N. Shapiro and Alexandra Shlapentokh were able to prove:
Hilbert's tenth problem is unsolvable for the ring of integers of any algebraic number field whose Galois group over the rationals is abelian. Shlapentokh and Thanases Pheidas (independently of one another) obtained the same result for algebraic number fields admitting exactly one pair of complex conjugate embeddings. The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open. Likewise, despite much interest, the problem for equations over the rationals remains open. Barry Mazur has conjectured that for any variety over the rationals, the topological closure over the reals of the set of solutions has only finitely many components. This conjecture implies that the integers are not Diophantine over the rationals and so if this conjecture is true a negative answer to Hilbert's Tenth Problem would require a different approach than that used for other rings.