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Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation $$ P(x_1,\dots,x_n) = 0 $$ where $P$ is a polynomial with integer coefficients. Do we have a standard definition/convention what exactly do we mean by "solving" such equation? If the number of solutions is finite this is clear: we need to list all the solutions and prove that there is no more. But what if the solution set is infinite? What description of this infinite set counts as "the solution" of the equation?

A trivial example is the equation $xy=0$, whose (integer) solution set is $(x,y)=(0,u)$ or $(x,y)=(u,0)$, where $u$ is an integer parameter. More generally, a polynomial family is the set of the form $(P_1(y_1,\dots,y_k),\dots,P_n(y_1,\dots,y_k))$, where $P_i$ are polynomials with integer coefficients, and $y_j$ are integer parameters. If we have represented the solution set of an equation as a finite union of polynomial families, then we certainly solved the equation. Unfortunately, for many equations, for example Pell equation $$ x^2 - 2y^2 = 1 $$ such representation of solution set is impossible. However, for this equation, it is easy to describe the solution set using recurrence relations.

Another standard example is finding rational solutions to equations that defines an elliptic curve. In this case the solution set forms a finitely generated group, and the set of generators of this group counts as a complete solution.

How to unify these examples in general definition? In each case, we have a complete description of the solution set, and this description can easily be used to enumerate/list all solutions. However, if we define that equation is solved if there is an algorithm that lists all its solutions, then all equations are trivially "solved" by an algorithm that just tries all candidate solutions in some order.

I am looking for a formal definition/convention in the form "We say that an equation is solved if and only if ... ".

As a specific example, what does it mean to find all integer solutions to the trivially-looking equations such as $$ x^2 + x = yz $$ or $$ x^2 + x + 1 = yz ? $$ We can define the divisor function $D(n)$ that, given a positive integer $n$, returns all its divisors. Then the solution set to the second equation is $(x,y,z)=(u, v, (u^2+u+1)/v)$, where $u \in {\mathbb Z}$ and $v \in D(u^2+u+1)$. But does this count as an adequate description of the solution set? Can we use the divisor function $D$? What exactly we can or cannot use?

Bogdan Grechuk
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    IMHO, I doubt that you can ask for a general "definition". It will depend on the geometry of the scheme. If the set of rational solutions is finite, e.g. the scheme has dimension 0 or it is a curve of genus at least 2, then you would wish to determine the set. If the solution set can be parametrised, e.g. a genus 0 curve, you want a full parametrisation. For an elliptic curve, you want a basis of the Mordell-Weil group etc. In higher dimensions it starts to be really hard to know what you are asking for. We often first focus on questions like are there ANY solutions? Are they dense?... – Chris Wuthrich Dec 08 '21 at 12:10
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    ... The best option I see is to say we solved it if we know as much as we can about the solution set. List subvarieties on which all the solutions lie, give a general construction (or even parametrisation) of some of them, ... – Chris Wuthrich Dec 08 '21 at 12:13
  • A solution should at least provide an algorithm for counting the number of solutions with all $x_i<n$ in time less than $n$. This is non-trivial! Perhaps the equations for which we have such algorithms match up well with the equations which we would say are solved. –  Dec 08 '21 at 12:14
  • One attempt at a semi-formal answer: to solve an equation means to provide some ("low-complexity") algorithm which on integer inputs enumerates the solutions. For instance, for an elliptic curve equation with solutions given by one generator $P$, we could give an algorithm which outputs coordinates of $nP$. – Wojowu Dec 08 '21 at 12:18
  • Surely the answer is just “a description of a recursively enumerable set” by Matiyasevich’s theorem? That is, to solve a Diophantine equation is to give a Turing machine which enumerates all its solutions. – Carl-Fredrik Nyberg Brodda Dec 08 '21 at 12:37
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    @Carl-FredrikNybergBrodda, by that definition we can trivially solve any Diophantine equation that has an infinite number of solutions, so I don’t think that definition captures what the OP is after. –  Dec 08 '21 at 13:14
  • For rational points, I think there is already no good definition of what it means to "solve" a K3 surface, assuming that K3 surface has infinitely many rational points. But if you want to construct a definition that is consistent and nontrivial, and not care that it matches ordinary mathematical usage of the term, you could mandate that we can "solve" an equation if and only if we have an algorithm that given a natural number $N$ and a residue class $x$ mod $N$ provably determines if there is a solution congruent to $x$ mod $N$. – Will Sawin Dec 08 '21 at 20:35
  • This gives the "right answer" in the Pell equation case: simply showing there are infinitely many solutions does not solve the equation, but expressing the solutions in terms of powers of one element of a number ring does. – Will Sawin Dec 08 '21 at 20:37
  • To "solve" an equation $P(x_1,…,x_n)=0$ means to ask if the corresponding formula $\varphi(x_1,\ldots, x_n,0)$ is satisfiable in a suitable domain, i.e. if $\exists x_1 \ldots x_n \varphi(x_1,\ldots, x_n,0)$ holds in that domain. – Mauro ALLEGRANZA Jan 12 '22 at 11:19
  • Will Sawin: with you definition, all quadratic equations in any number of variables (not only Pell equation) are automatically solved: substitution $Nx_i+r_i$ instead of every variable $x_i$ keeps the equation quadratic, and then apply a well-known algorithm of Grunewald and Sigel to check whether the resulting equation has any integer solution. – Bogdan Grechuk Jan 28 '22 at 18:37

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