Talk:Rational point

Content concerns
This article is incomprehensible, and it appears to be wrong. First off, K is not introduced in any way - ok it's a field. It looks to me like the article defines rational points as those points that are solutions to a polynomial. I thought, for instance, that if your polynomial ring is, say, Z[x,y] - then rational points would be solutions x, y in Z, not in I dunno the field of fractions of Z. Also, the article unfortunately plunges into a bunch of theorem name-dropping without addressing the topic itself (this seems common on math pages these days). Please make an attempt to explain what rational points are rather than try to impress us with your encyclopedic knowledge. I'm no expert but I'm trying to find things out.

Ok, I see that the definition is correct - or would be if it were mentioned what k is, and presented with more of an introduction. Also the possibility of rational points over a field extension of the base field could be mentioned. It's just frustrating how many good math articles have morphed into these mathematical pissing contests that inform nobody who isn't already informed. Who are you trying to impress, anonymous internet entities? God? — Preceding unsigned comment added by MatthewCushman (talk • contribs) 17:16, 15 January 2011 (UTC)
 * I agree.-Rich Peterson76.218.104.120 (talk) 04:14, 2 January 2012 (UTC)
 * I think you don't need to be so aggressive. It is not always easy to explain everything. Since you figured out what k is, just put it in the article. That is how WP works. Liu (talk) 23:41, 3 January 2012 (UTC)
 * MatthewCushman has made 3 edits including the two above. I think you're both talking to a ghost. If someone actually understands anything on this or the article page (I sure don't), maybe they should address any and all concerns there may be relating to the article. I wouldn't know if the math was upside down and back to front!  fredgandt  23:48, 3 January 2012 (UTC)

Density
It would be helpful to comment on the density of rational points on a variety. Under what conditions does this hold, and which sense of "dense"? Tkuvho (talk) 12:59, 5 January 2012 (UTC)


 * "Most" algebraic varieties have very few rational points, as the case of $$x^n+y^n=z^n$$ famously illustrates.  Sławomir Biały  (talk) 13:11, 5 January 2012 (UTC)


 * True, but in certain Diophantine problems it does, see Weil's book. Tkuvho (talk) 13:49, 5 January 2012 (UTC)


 * It may indeed be true that in some sense 'Most' algebraic varieties have very few rational points. But with two vague quantities here ( 'Most'  and very few) the sentence is virtually meaningless.  ALSO: What exactly is the difference between the word Most and "Most", as in the claim by Sławomir Biały ?  FINALLY: The case of $$x^n+y^n=z^n$$ does nothing to illustrate that  'Most' algebraic varieties have very few rational points, since it is only one case (for each degree).Daqu (talk) 06:09, 17 June 2012 (UTC)

title doesn't fit
A better title for what is written here would be "Rational points on curves and varieties".198.189.194.129 (talk) 20:01, 11 January 2012 (UTC)


 * Since there is no article Rational point other than this one, and so many other articles link to this for the description, I'd recommend that the title remains the same. If there is a more specific subject covered in the article, than the name suggests, it is the content that should be changed. The subject should be approached from an angle that first describes the whole subject, then how that applies to sub-subjects. So if at present the article describes Rational points on curves and varieties, its content should be rewritten to account for all else a rational point is, and Rational points on curves and varieties should become a section. There is no deadline though. Face-smile.svg  fredgandt  01:37, 25 January 2012 (UTC)

integral points
What with all the emphasis here on rational or k-rational points on curves etc, it would be good to mention rational integer points on curves, and the k-integer generalizations.--Richard Peterson198.189.194.129 (talk) 20:06, 11 January 2012 (UTC)
 * I know you already are going for it, but I just thought I'd say "Go for it!" anyway.  fredgandt  01:40, 25 January 2012 (UTC)
 * thanks for the encouragement.198.189.194.129 (talk) 00:21, 27 January 2012 (UTC)

Is this article still a stub?
For info - Stub

As I see it, a stub is an article that is sorely lacking a full coverage of the subject. Now this article has been bulked out to include lots of words and numbers that may possibly mean something, could we consider it to be a full article? It may be short (but not terribly short), but if the subject is properly covered, I wouldn't call it a stub. Other interpretations may consider that the length is the governing factor. Hmm?  fredgandt  00:32, 25 January 2012 (UTC)
 * yes, I agree, not a stub. Also, I recently discovered that the bulk of the article, on rational points on varieties, is the subject of the article Diophantine geometry.--Rich P.198.189.194.129 (talk) 00:32, 26 January 2012 (UTC)


 * Well you're the math expert. If you think the Dolphin geography article covers the ground (of Rational points) already, perhaps this could become a redirect to it, after merging the content into the Dolph Lundgren geology article. Since the term Rational point is used quite frequently in other math articles, I think the term has enough significance to hold its head up and be an article in its own right. So, another possibility would be to merge the content describing rational points from the Deuterium geopolitical article, into this one, and provide a main at the Duodenum geothermal section dealing most heavily with the topic. Basically, the best solution (to avoid doubling/halving the effort/coverage) would result in the fewest redirects (which cause confusion) and the best coverage of the subject. If a reader searches for Rational point and lands at Delphinium geostationary, they will perhaps be mildly astonished, but if they are searching for and reading general mathematics and wish to know more about rational points, they would possibly be less perturbed by the need to visit another article dealing explicitly with that subject. The possibilities are endless. There can be only one! What d'ya reckon?  f<i style="color:#0dd;font-size:10px;">red</i>g<i style="color:#0dd;font-size:10px;">andt</i>  22:39, 26 January 2012 (UTC)
 * P.S. Congratulations for getting a name! It suits you. I see you've had it a while, and just don't always use it. Each to their own.  f<i style="color:#0dd;font-size:10px;">red</i>g<i style="color:#0dd;font-size:10px;">andt</i>  22:39, 26 January 2012 (UTC)


 * Having just looked at Diophantine geometry (jokes aside); the two articles are quite definitely not alike. Not mirroring each other, but complementing each other. Since Diophantine equation only has one mention of Rational points, it's also fair to say that article doesn't do the subject full justice. I'd say, this article really does deserve to remain the home of this knowledge, and should not become a redirect. No way.  f<i style="color:#0dd;font-size:10px;">red</i>g<i style="color:#0dd;font-size:10px;">andt</i>  22:56, 26 January 2012 (UTC)
 * Agreed.--Richard Peterson198.189.194.129 (talk) 00:21, 27 January 2012 (UTC)

Confusing definitions
The article is confusing in the most crucial way: how it defines its subject.

Just after defining rational points on an algebraic variety as those points with rational coordinates, it throw in a "special case" of integer points (whose coordinates are of course integers).

But this is not a special case of the definition of rational points in the way that definition is phrased. It is not the set of points on the variety whose coordinates are rational. (Ditto for the term K-rational and the algebraic integers of K.) These are, of course, subsets of the rational points. But what subsets are permissible? Any subsets whatsoever? And if they're (algebraic) integer points, why are they called rational points?

ALSO: even more confusingly, the second paragraph of the section "Rational or K-rational points on algebraic varieties" reads:

"The Weil conjectures concern the distribution of rational points on varieties over finite fields."

Once again taking the article literally, there is no rational number in a finite field. So the article should be corrected to reflect this.Daqu (talk) 04:10, 8 March 2012 (UTC)
 * I made a change to the 'Weil conjectures' sentence that hopefully clarifies it, you're right to criticize it. I'm still thinking about your points other than that one. At the moment, however rational your points may be, I don't get them. {:->  Rich Peterson76.218.104.120 (talk) 22:02, 6 June 2012 (UTC)


 * Sorry to have been unclear. But just now looking at the introduction, I found a sentence:


 * "A special case of rational points are integer points . . .."


 * with disagreement in number between the subject and the verb. I think that earlier I was confused about whether the introduction was referring to the set of rational points, or individual ones. If it had been the set of them, then the set of integer points is not a "special case"; it is just a subset. It is better to call an integer point an example of a rational point.


 * In any case, these sentences in the introduction remain confusing :


 * "In number theory, a rational point is a point in space each of whose coordinates are rational; that is, the coordinates of the point are elements of the field of rational numbers, as well as being elements of larger fields that contain the rational numbers, such as the real numbers and the complex numbers."


 * and


 * "On the other hand, more generally, a K-rational point is a point in a space where each coordinate of the point belongs to the field K, as well as being elements of larger fields containing the field K."


 * The problem is that the phrases that begin "as well as being elements of larger fields" must be extremely puzzling to most readers. Of course a point of any set X is also a point of any set that contains X as a subset.  So why mention it in these cases?


 * I recognize that in algebraic geometry one is often concerned with questions like which points of a complex variety are also rational points, and I believe that is the reason these phrases are included. But to most readers, their inclusion will just be confusing . . . if they are in the introduction.    Later in the article it might make sense to discuss rational points in larger fields.  Or, this may be appropriate in the introduction if the situation is explained clearly. This would require a major change of wording.


 * Clear wording would start by mentioning a space with coordinates (not all spaces are such!) that are defined over the real or complex field. And then refer to the issue of whether some points of the space have all their coordinates in the rational numbers. Later in the article, the more general situation of [starting with a space defined over one field and then asking about those points of the space whose coordinates lie in a subfield] could be mentioned. But a generalization of the subject of an article does not belong in the introduction.Daqu (talk) 16:28, 15 June 2012 (UTC)

Rational points on varieties not defined over the rational numbers
If V is an algebraic variety not defined over Q, one can't talk about rational points V(Q). Here is a reason. Suppose V is defined over a big field K, then any variety over K isomorphic to V should have the same Q-points (because one can't distinguish two K-isomorphic varieties). But clearly, a K-isomorphism can changes the Q-points (defined in an intuitive way). For example, the curve y=x+a with a irrational, is isomorphic (over the smallest field it is defined, Q(a)) to the affine line, and the latter has full of Q-points while the former doesn't have any Q-point. -- unsigned comment added 18:45, 9 June 2012‎ by Uni.Liu (talk)


 * A simpler way of proving this is: Assume we have a morphism Spec Q &rarr; X. The image of this morphism is a point of X.  This point lies in some affine chart Spec A, and the morphism corresponds to a homomorphism A &rarr; Q.  If X is a K-variety, then A is a K-algebra, and hence we deduce the existence of a non-zero homomorphism K &rarr; Q; this implies K = Q.  Ozob (talk) 01:28, 16 June 2012 (UTC)

violations of wiki policy
this article is apparently in violation (of varying degrees) of two fundamental wiki policies.

1. Make technical articles understandable

2. No original research


 * Would you be more specific? I don't see any original research, and the article makes sense to me.  Ozob (talk) 03:19, 31 July 2013 (UTC)

Imaginary point: proposed merge
It has been proposed that Imaginary point should be merged here.


 * Strong oppose. The nature of imaginary/complex and rational coordinates are entirely different. I'd suggest that the Imaginary point stub be merged with Complex projective space. &mdash; Cheers, Steelpillow (Talk) 12:30, 5 February 2015 (UTC)
 * I disagree with your assessment; imaginary point describes $$\mathbf{CP}^2 \setminus \mathbf{RP}^2$$, meaning that from a scheme-theoretic perspective, it's the $$\mathbf{C}$$-rational points of $$\mathbf{P}^2$$ which aren't $$\mathbf{R}$$-rational. That fits in with the general concept of rationality discussed at the tail end of this article (though I agree it doesn't fit very well with the discussion of $$\mathbf{Q}$$-rationality that dominates the article).  Having said that, I think I would only be comfortable with the merge if this article contained more information on K-rational points for general fields K.  Ozob (talk) 14:50, 5 February 2015 (UTC)
 * Much of this is above my head. I suppose you are going to tell me that an "imaginary point" is not just a complex point whose real component is zero? &mdash; Cheers, Steelpillow (Talk) 18:43, 5 February 2015 (UTC)
 * Not according to the linked article. The "imaginary points" of that article are more like $$\mathbf{C} \setminus \mathbf{R}$$.  Note that this also disagrees with Taku's proposed definition below.
 * I don't like the definition in the imaginary point article (I would rather use one like Taku's). And the article has no references, so I find myself doubting whether it's a concept that appears in the literature.  It might be better just to redirect to imaginary number.  Ozob (talk) 02:07, 6 February 2015 (UTC)
 * It seems I didn't actually understand the context behind the article, which is (classical) projective geometry. (Of course, "in geometry" is not very helpful in establishing the context, but anyway.) The merger is not a good idea after all. -- Taku (talk) 19:43, 8 February 2015 (UTC)


 * Comment. Ozob has given my rationale for the merger. It is problematic that the article is about both coordinates with rational numbers and a functorial point that is often called a rational point (namely one from Spec of a field or valuation ring). The matter of merger should however be somehow independent of the quality and state of the article but should depend on our future plan of the article. I'm assuming that the last section of the article will be vastly expanded in the future. In particular, the article should discuss specialization map. -- Taku (talk) 18:57, 5 February 2015 (UTC)
 * So it seems that an "imaginary point" is some abstruse thing that I have never heard of. Would it be appropriate to add a disambig link along the lines of:
 * &mdash; Cheers, Steelpillow (Talk) 19:07, 5 February 2015 (UTC)
 * Yeah, so I think I would define an imaginary point to be a complex point that is fixed by (1/2)(id - complex conjugation). This definition would work whether a point is functoral or set-theoretic. So I don't think there is a need for a disambig. We simply have to explain why a functorial point is a generalization of a set-theoretic point. -- Taku (talk) 19:20, 5 February 2015 (UTC)
 * Yeah, so I think I would define an imaginary point to be a complex point that is fixed by (1/2)(id - complex conjugation). This definition would work whether a point is functoral or set-theoretic. So I don't think there is a need for a disambig. We simply have to explain why a functorial point is a generalization of a set-theoretic point. -- Taku (talk) 19:20, 5 February 2015 (UTC)


 * Strong ppose — As per Steelpillow. An element in a vector space of which the components with respect to some (privileged) basis (as a rational point is defined in this article) is not even remotely the same thing as a point in a projective geometry as defined in Imaginary point, for which the components expressed in terms of some special homogeneous basis happens to satisfy a ratiometric criterion. —Quondum 01:36, 8 February 2015 (UTC)
 * Actually a rational point is a coordinate-free notion; for example, a real point is a point fixed by the action of complex conjugation and such an action need not be tied to coordinates (but is tied to, I suppose, a choice of complex structure; see for instance complex vector bundle). But I now have better understanding of context (thanks to efforts by David and you) and I now agree the merger is not a way for us to go. -- Taku (talk)

Is the definition correct? Seems sloppy to me...
The article first lets $$X$$ be an algebraic variety in affine $$n$$-space $$A^n$$ over a field $$k$$, defined by polynomials $$(f_j)_{j \in \{1,\ldots,r\}}$$. It then defines a $$k$$-rational point of $$X$$ to be an element of $$(a_1,\ldots,a_n) \in k^n$$ such that $$f_j(a_1,\ldots,a_n)=0$$ for all $$j$$. But surely under this definition, every point of $$X$$ is a $$k$$-rational point? Is the definition perhaps meant to define a $$K$$-rational point of a $$k$$-variety (with $$K \neq k$$?) Iteraf (talk) 17:07, 11 June 2018 (UTC)
 * Good point. I have fixed the definition, and added an example. D.Lazard (talk) 19:32, 11 June 2018 (UTC)

Subcategorization
regarding, notice that Category:Diophantine geometry is a subcategory of Category:Algebraic geometry. By WP:SUBCAT, we shouldn't leave the article in the parent category as well. The article is automatically in the parent category by being in the subcategory! — MarkH21talk 11:14, 13 January 2020 (UTC)
 * Ok. I had a view of Diophantine geometry that is less broad than what is presented in the artice, essentially similar to geometry of numbers. I'll revert myself. D.Lazard (talk) 12:30, 13 January 2020 (UTC)

Names of fields in the definition
Maybe it's because I read too quickly, but I was very confused by "k" being a field. I'm used to fields being capital F, often in blackboard bold, and lower case "k" being "some variable smaller than n". (e.g. k out of n in combinatorics). Like, the definition makes sense, but I found the names used a little counterinituitive. Do with this feedback what you want :) Just my 2 cents. — Preceding unsigned comment added by 2001:9E8:6300:8400:26D9:40C8:CBEF:1110 (talk) 10:03, 4 April 2022 (UTC)
 * Please, do not confuse habits of some teachers with conventions used outside schools. It is extremely rare that a symbol used to denote an object introduced by "let [symbol] be ..." results of a convention. It is true that $k$ is commonly used for integers, but it may have many other meanings, and when it is used, the user must say what is denoted by it. This is the purpose of the introducing sentence "Let $k$ be a field". For unspecified fieds, the commonly used symbols are $F$ (abbreviation of "field"), $k$ and $K$ (abbreviations of the German word "Korper"), and $$\mathbb F$$ is discouraged by most authors. In the context of this article, one needs two fields, one being larger than the other. It is thus natural and very common to name them $k$ and $K$, since other notations would be either less mnemonic (such as $F$ and $G$), or more complicate (such as $K$ and $$\overline K$$). D.Lazard (talk) 11:30, 4 April 2022 (UTC)