Timeline for What does it mean to solve an equation?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 28, 2022 at 18:37 | comment | added | Bogdan Grechuk | 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. | |
| Jan 12, 2022 at 11:19 | comment | added | Mauro ALLEGRANZA | 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. | |
| Dec 8, 2021 at 20:37 | comment | added | Will Sawin | 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. | |
| Dec 8, 2021 at 20:35 | comment | added | Will Sawin | 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 $N$. | |
| Dec 8, 2021 at 13:14 | comment | added | user44143 | @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 8, 2021 at 12:37 | comment | added | Carl-Fredrik Nyberg Brodda | 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. | |
| Dec 8, 2021 at 12:18 | comment | added | Wojowu | 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 $nP$. | |
| Dec 8, 2021 at 12:14 | comment | added | user44143 | 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 8, 2021 at 12:13 | comment | added | Chris Wuthrich | ... 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, ... | |
| Dec 8, 2021 at 12:10 | comment | added | Chris Wuthrich | 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?... | |
| Dec 8, 2021 at 11:00 | history | asked | Bogdan Grechuk | CC BY-SA 4.0 |