Talk:Hilbert's tenth problem

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The article claims:

The equation

$p(x_{1},\ldots ,x_{k})=0$

where $p$ is a polynomial of degree $d$ is solvable in rational numbers if and only if

$(z+1)^{d}\;p\left({\frac {x_{1}}{z+1}},\ldots ,{\frac {x_{k}}{z+1}}\right)=0$

is solvable in natural numbers.

This cannot be true. x+1=0 is solvable in rational numbers, but x+z+1=0 is not solvable in natural numbers. 141.35.26.61 03:41, 21 January 2007 (UTC)

I suspect the intent was to solve the original equation over the positive rationals. But I've changed "naturals" to "integers" in the article. Ben Standeven 05:14, 7 April 2007 (UTC)

It didn't work that way either; in your version, the case z=0 caused problems. I think I've fixed it now. 141.35.26.61 12:28, 10 April 2007 (UTC)

I removed this part that makes no sense : a) the term "non negative" appears randomly while we are talking about rings, and has nothing to do with the discussion which seems to be integers vs rational numbers b) the idea to change a rational equation in the integer equation $(z+1)^{d}\;p\left({\frac {x_{1}}{z+1}},\ldots ,{\frac {x_{k}}{z+1}}\right)=0$ only shows that if there was an algorithm for problems over Z there would also be an algorithm for problems over Q. Since there is no algorithm for problems over Z, what is the point of mentioning this ? c) This may confuse the reader to think that Hilbert's problem over Q is solved, while it is not cf for example http://math.mit.edu/~poonen/papers/aws2003.pdf Section 12 --194.214.160.170 (talk) 16:04, 4 November 2015 (UTC)

Flaw in Article

There is no meaning for $p(a,x_{1},\ldots ,x_{k})=0$ A student that knows about polinomials may understand the article until (s)he finds such a notation with no reference to its meaning and much less its discussion of "parameters"

"...with integer coefficients such that the set of values of a for which the equation

   p(a,x_1,\ldots,x_n)=0

has solutions in natural numbers is not computable. So, not only is there no general algorithm for testing Diophantine equations for solvability, even for this one parameter family of equations, there is no algorithm ..."

For lack of refences (s)he simply gets lost.

If the article is just for those who know about it the What is the purpose of the article? —Preceding unsigned comment added by Siniestra (talk • contribs) 22:17, 21 November 2008 (UTC)

I've added a sentence to the intro, explaining this notation and the concept of Diophantine equations. That seems to be a prerequisite to understanding this article, so I don't think we need much detail.Ben Standeven (talk) 15:26, 31 March 2009 (UTC)

EXP formula

What is the Diophantine equation of EXP? --77.125.2.128 (talk) 14:46, 14 September 2009 (UTC)

polynomial

From the article:

There exists a polynomial $p(a,n,x_{1},\ldots ,x_{k})$ such that, given any Diophantine set $S$ there is a number $n_{0}$ such that

$S=\{\,a\mid \exists x_{1},\ldots ,x_{k}[p(a,n_{0},x_{1},\ldots ,x_{k})=0]\,\}.$

The question: had someone wrote down such a polynomial? --84.229.245.201 (talk) 08:48, 28 December 2011 (UTC)

Extremely poor writing

The section Formulation begins as follows:

"The words "process" and "finite number of operations" have been taken to mean that Hilbert was asking for an algorithm. The term "rational integer" simply refers to the integers, positive, negative or zero: 0, ±1, ±2, ... . So Hilbert was asking for a general algorithm to decide whether a given polynomial Diophantine equation with integer coefficients has a solution in integers. Such an equation has the following form:

$p(x_{1},x_{2},\ldots ,x_{n})=0.$

The answer to the problem is now known to be in the negative: no such general algorithm can exist."

No, the equation does not have the "following form" when the expression

$p(x_{1},x_{2},\ldots ,x_{n})$

is left unqualified.

It also does not have the "following form" at all, since a "Diophantine equation" does not necessarily have a 0 on one side of the equation.

It is equivalent to an equation of the form displayed when

$p(x_{1},x_{2},\ldots ,x_{n})$

is described as a polynomial with integer coefficients in the variables

$x_{1},x_{2},\ldots ,x_{n}.$

Daqu (talk) 21:37, 7 April 2015 (UTC)

For how many unknowns is there a known existence-deciding algorithm?

The section Hilbert's tenth problem#Further results says Zhi Wei Sun showed that the problem for integers is unsolvable even for equations with no more than 11 unknowns.

1. Is there a value n > 1 such that an algorithm is known for deciding the existence of solutions for problems with no more than n unknowns?

2. If algorithms are known for problems with 2, or 3, unknowns, how do they work?

(Trivially an algorithm for deciding existence of a solution is known when n=1: the rational root test.)

I think the article should answer the first of these, either affirmatively or negatively, and if affirmatively then it should answer the second question as well, at least in summary form. Loraof (talk) 20:31, 9 June 2017 (UTC)

More references, please!

The article provides a good general introduction to the area, and a fair summary of some of the more important results achieved by those who have attacked Hilbert's Tenth Problem. However, the article sorely needs more references, for (at least) these reasons:

1. It's WP policy to quote reliable sources (RS) in support of non-obvious assertions.
2. More importantly, readers given a source reference can check the facts and derivations for themselves.
3. Further, curious readers can use a well-referenced article as a springboard for further exploration and learning.

Please give appropriate references if you have them to hand! yoyo (talk) 14:55, 15 June 2017 (UTC)

Semidecidability

Some mention should be made of whether or not diophontine equations can be semidecided for integer solutions. It looks to me like they should be eventually decidable IFF there is an integer solution. Simply trying out different values should eventually yield a solution if one exists. Is there any reason why this would not be the case for certain types of diophontine equations? Comiscuous (talk) 20:51, 16 December 2021 (UTC)

Section Hilbert's_tenth_problem#Recursively_enumerable_sets already states "It is evident that Diophantine sets are recursively enumerable. This is because one can arrange all possible tuples of values of the unknowns in a sequence and then, for a given value of the parameter(s), test these tuples, one after another, to see whether they are solutions of the corresponding equation." I added "a.k.a. semi-decidable", guessing that was the string you searched for. - Jochen Burghardt (talk) 08:10, 17 December 2021 (UTC)