Template:Did you know nominations/Height function

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: promoted by Yoninah (talk) 19:14, 21 April 2019 (UTC)

Height function

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... that height functions allow mathematicians to count and distinguish objects such as rational points of algebraic varieties? Source: Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. pp. 43–67. ISBN 3-540-61223-8. Zbl 0869.11051.
- ALT1:... that height functions were central ideas in the proofs of the Mordell–Weil and Faltings's theorems about rational points of algebraic varieties? Source: Weil, André (1929). "L'arithmétique sur les courbes algébriques". Acta Mathematica. 52 (1): 281–315. doi:10.1007/BF02592688. MR 1555278. and Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae (in German). 73 (3): 349–366. doi:10.1007/BF01388432. MR 0718935.
- ALT2:... that the unsolved Manin and Vojta conjectures about height functions of rational points of algebraic varieties have far-reaching implications in Diophantine approximation and arithmetic geometry? Source: Vojta, Paul (1987). Diophantine approximations and value distribution theory. Lecture Notes in Mathematics. Vol. 1239. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0072989. ISBN 978-3-540-17551-3. MR 0883451. Zbl 0609.14011. and Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Annals of Mathematics. 123 (3): 549–576. doi:10.2307/2944319. MR 1109353.

Converted from a redirect by MarkH21 (talk). Self-nominated at 10:52, 12 April 2019 (UTC).

Comment: this was nominated eight days late by a first-time DYK nominator; as such, I'd like to suggest that we use our discretion, as we typically do with newcomers, to allow the nomination to proceed normally as if it had been nominated seven days after work began rather than fifteen. For future reference, it's better to nominate on time, within seven days of creation/conversion/expansion and then finish building the article. BlueMoonset (talk) 16:04, 18 April 2019 (UTC)

I'm agreed that it's fine to accept this nomination. I'll review it. — Bilorv (he/him) _(talk) 11:40, 21 April 2019 (UTC)

Sorry about that! I recognized that there is a 7 day limit but I also saw WP:DYKSG#D9 and I figured that the backlog isn't too great. I'll work on earlier nominations next time :) — MarkH21 (talk) 18:55, 21 April 2019 (UTC)

General: Article is new enough and long enough
New enough: Long enough:

Policy: Article is sourced, neutral, and free of copyright problems
Adequate sourcing: Neutral: Free of copyright violations, plagiarism, and close paraphrasing:

Hook: Hook has been verified by provided inline citation
Cited: Interesting:

QPQ: None required.

Overall: ALT0 approved, as I reckon it's the simplest and most interesting of the hooks. I've got a couple of suggestions, but they are not necessary to meeting DYK criteria, just ideas to improve the article. An inline citation for the Néron–Tate height section would be good (it looks like the first reference in Néron–Tate height, to Néron's original paper, would suffice). The lead of mathematical articles are generally intended to be a level or two more accessible than the body, so in this case I think it would be helpful to define in the lead the height of a polynomial and give an explicit example (e.g. $3-4x+2x^{2}$ has height 4). It could also be helpful to give an example of height on Q—for instance, if I'm understanding things correctly, ${\frac {20}{50}}$ has height 5 as its simplest form is ${\frac {2}{5}}$ , and the rational numbers with height $\leq 2$ are $-2,-1,-{\frac {1}{2}},0,{\frac {1}{2}},1,2$ . — Bilorv (he/him) _(talk) 11:40, 21 April 2019 (UTC)

Thanks for taking the time to review and provide feedback! I agree with your suggestions, so I added two Néron–Tate height references (including the one you mentioned) and added a basic example for naive height (the most basic and influential height function) to the lead and its corresponding section. Your understanding is correct!