Mathematics Asked on November 17, 2020

Hilbert’s tenth problem is the problem to determine whether a given multivariate polyomial with integer coefficients has an integer solution. It is well known that this problem is undecidable and that it is decidable in the linear case. In the quadratic case (degree $2$) , the case with $2$ variables is decidable.

Is the case of degree $2$ decidable ? And if yes, can we always find the complete solution set ?

I ask this because many diophantine equations turn out to be solvable despite of the negative result of the problem. Wikipedia gives upper bounds for the degree and for the number of variables sufficient to make the problem undecidable, but I nowhere found a classification of the cases known to be solvable , apart from the elliptic curves and the cases I mentioned above.

I quote from page $1$ of the same notes:

A

Diophantine problemover $mathbb{Q}$ is concerned with the solutions either in $mathbb{Q}$ or in $mathbb{Z}$ of a finite system of polynomial equations $$F_i(X_1, ldots , X_n) = 0 hspace{0.5in} (1 leq i leq m) hspace{1.0in} (1)$$ with coefficients in $mathbb{Q}$. Without loss of generality we can obviously require the coefficients to be in $mathbb{Z}$. A system $(1)$ is also called a system ofDiophantine equations. Often one will be interested in a family of such problems rather than a single one; in this case one requires the coefficients of the $F_i$ to lie in some $mathbb{Q}(c_1, ldots , c_r)$, and one obtains an individual problem by giving the $c_j$ values in $mathbb{Q}$. Again one can get rid of denominators. Some of the most obvious questions to ask about such a family are:

(A)Is there an algorithm which will determine, for each assigned set of values of the $c_j$, whether the corresponding Diophantine problem has solutions, either in $mathbb{Z}$ or in $mathbb{Q}$?

(B)For values of the $c_j$ for which the system is soluble, is there an algorithm for exhibiting a solution?

For individual members of such a family, it is also natural to ask:

(C)Can we describe the set of all solutions, or even its structure?

I quote from pages $13$ to $14$ of these progress notes on Diophantine equations:

The most important invariant of a curve is its

genus. In the language of algebraic geometry over $mathbb{C}$, curves of genus $0$ are calledrational, $ldots$ A canonical divisor on a curve $Gamma$ of genus $0$ has degree $−2$; hence by the Riemann-Roch Theorem $Gamma$ is birationally equivalent over the ground field to aconic. The Hasse principle holds for conics, and therefore for all curves of genus $0$; this gives a complete answer to Question(A)at the beginning of these notes. But it does not give an answer to Question(B). Over $mathbb{Q}$, a very simple answer to Question(B)is as follows:

Theorem 1Let $a_0, a_1, a_2$ be nonzero elements of $mathbb{Z}$. If the equation $$a_0 {X_0}^2 + a_1 {X_1}^2 + a_2 {X_2}^2 = 0$$ is soluble in $mathbb{Z}$, then it has a solution for which each $a_i {X_i}^2$ is absolutely bounded by $|a_0 a_1 a_2|$.

Siegel has given an answer to Question

(B)over arbitrary algebraic number fields, and Raghavan has generalized Siegel’s work to quadratic forms in more variables.

The knowledge of one rational point on $Gamma$ enables us to transform $Gamma$ birationally into a line; so there is a parametric solution which gives explicitly all the points on $Gamma$ defined over the ground field. This answers Question

(C).

Correct answer by Arnie Bebita-Dris on November 17, 2020

The OP asked for further inputs on the two-variable case of Hilbert's Tenth Problem.

One can check out the discussion and answers to this closely related MO question: Connection between the two-variable case of Hilbert's Tenth Problem and Roth's Theorem..

I quote Felipe Voloch:

"(answer) $ldots$ The case of diophantine equation of two variables is

generally believedto be decidable. Poonen has a couple of expository articles on this (check his web page) that you might find useful."

"(comment)

The state of the art today is that two variable problems can in practice be solved and systematically are.We just can't yet prove that we will always succeed. Look, e.g., at the papers of M. Stoll and his collaborators."

Answered by Arnie Bebita-Dris on November 17, 2020

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