Talk:Zero of a function

Root vs. Zero

I prefer 'root' for 'root of an equation' and `zero' for 'zero of function'. Lhf 02:04, 29 May 2007 (UTC)

Indeed! Equations have roots, while polynomials (and other functions) have zeroes. The article should be renamed. 78.73.89.202 (talk) 08:51, 2 July 2010 (UTC)

I agree. BUT THE DAMN CAPION! Jeez... —Preceding unsigned comment added by 69.232.201.135 (talk) 09:14, 18 September 2010 (UTC)

Y-intercept

Y-intercept would be a great addition. --Abdull 15:28, 31 May 2006 (UTC)

The y- intercept of f is simply f(0). It really has no significance that I can see. It might be useful to add a warning no to confuse the two, which I'll add. He Who Is 17:50, 10 June 2006 (UTC)

P and Q

I think information about P and Q should be added here. P being factors of $a_{0}$ and q being factors of $a_{1}$.

Then p/q=possible roots --Misantropo 00:30, 18 July 2006 (UTC)

The fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities

In the claim "the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities" should the word "non-zero" be added before "polynomial"? Jayme 12:16, 25 May 2007 (UTC)

I'm not an expert by any means, but I'm pretty sure zero isn't a polynomial. —Keakealani·?·!·@ 23:05, 27 May 2007 (UTC)

I believe that the usual definitions of polynomial include the zero polynomial. Maybe this is more clear if we regard 0 as a₀ + a₁x + a₂x² + ... with a_n = 0 for all n. Jayme 23:41, 2 June 2007 (UTC)

On Gallian's "Contemporary Abstract Alegbra" 2006 (page 293) the degree of a polynomial in the form a_n*x^n +...+ a_0 has leading coefficient of a_n and degree n iff a_n != 0. A constant would thus have degree 1, except for the zero polynomial which would not have a degree. Inclusion of the fundamental theorem of algebra and even a justification for imaginary roots using the cubic equation x^3=3px+2q is something that I think would be important to the entry. —Preceding unsigned comment added by Mrchapel0203 (talk • contribs) 05:48, 5 September 2008 (UTC)

Yes, the word "non-zero" should be added! The exists a zero polynomial and its degree is either not defined or negative.

zeros vs root

zeros and roots are different fools, fix it! root: (x-a), where the a is the zero —Preceding unsigned comment added by 203.24.97.11 (talk) 11:43, 1 November 2007 (UTC)

Whoever taught you that should lose their teaching certificate. (ESkog)^(Talk) 13:08, 1 November 2007 (UTC)

Reverted Change: Applications

I made the following edit recently to add an "Applications" section:

Direct applications of being able to calculate the zero of functions is in solving a System of linear equations, since f(x) - g(x) = 0 is the equivalent of saying f(x) = g(x). This allows you to find the points where the function graphs intersect. As a trivial example, if you have two items each with their own function describing cost over time, one with increasing costs and one with decreasing costs, it would be possible to figure out when the costs are equal and will reverse roles. More generally, this concept is core to the understanding of Linear algebra and in polynomial systems, System of polynomial equations.

It was reverted for being confusing and containing mathematical errors. While I understand why one might revert such changes, I do think it is worth having an "applications" or "related concepts" section. I appreciate the high quality of the math articles that exist in Wikipedia, but it is frustrating to me that such an important concept entirely lacks an applied or related section. I am a math enthusiast, although obviously not at all a very advanced one, but I mostly take interest in understanding concepts + some additional useful contexts. That information is most valuable to me.

If any people interested in education as well as general information would be willing to create a more accurate and less confusing "applications", "related", or "examples" section, I would certainly appreciate it, as I'm sure there are others who have gone through years of math education and still don't understand "why" we care about zeros / roots of functions, and many of us use Wikipedia as a learning resource for mathematics.

If you have any alternative proposals or suggestions, or any comments, that is of course fine as well.

Pritchard 13:49, 7 June 2015 (UTC)

I am also wondering if page maintainers would find more granular sections (applications in linear algebra / systems of linear equations, finding the solution to a specific problem example) acceptable. I am asking only to try and make these pages both correct and useful to more entry-level mathematics enthusiasts as well.

Pritchard 13:57, 7 June 2015 (UTC)

I generally agree with your comments. However, in this case this seems unnecessary, because of the new section that I have just added to the article: For more information on the subject, it suffices to consult the article Equation. D.Lazard (talk) 16:58, 7 June 2015 (UTC)

That seems fine with me! Thank you for your ongoing contributions. Pritchard 15:24, 8 June 2015 (UTC)

Merge Zero (complex analysis)

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

The result of this discussion was to merge Zero (complex analysis) and Pole (complex analysis) into the new article Zeros and poles. D.Lazard (talk) 21:29, 14 January 2018 (UTC)

---

The article Zero (complex analysis) describes the same as this article, with only a few differences:

1. it treats real zeros only in passing
2. it defines multiplicity and simple zero
3. it mentions that zeros are isolated for holomorphic functions

Since the differences are so few, I suggest merging the two. Since this article is linked 259 times (counting indirect links via redirects, but not the redirects themselves), and the other article only 59 times, and since I believe this title to be more likely entered in a search, I suggest merging the other article into this one. Any objections? — Sebastian 10:37, 18 December 2017 (UTC)

• Weak support or weak oppose. I agree with the rationale of the nominator. Nevertheless, Zero (complex analysis) is a stub that contains almost nothing about what it should contain. In fact, the importance of zeros in complexes analysis lies in the duality between zeros and poles that appears in Riemann–Roch theorem, and in the fact that the divisor of zeros and poles of a global meromorphic function has degree zero. Therefore my suggestion is to merge Zero (complex analysis) and Pole (complex analysis) into Zeros and poles, and to expand the merged article by the main facts on zeros and poles that are at the basis of complex analysis (including those suggested above). D.Lazard (talk) 11:28, 18 December 2017 (UTC)

  Thank you for your justification for a Zeros and poles article; that sounds indeed like a topic that deserves its own article. The only thing that bugs me is that that would destroy the symmetry in our article structure, which we currently enjoy e.g. in such places as Pole–zero plot#Poles and zeros. If we merged the poles article into a Zeros and poles article, what should the word "poles" link to, if we want to keep the symmetry of the two sentences? — Sebastian 14:23, 18 December 2017 (UTC)

If my suggestion would accepted, Zero (complex analysis) and Pole (complex analysis) should both redirect to Zeros and poles, and this new article should link Zero of a function in the first lines of the lead or in a hatnote. The alphabetic order would suggest Poles and zeros, but I prefer Zeros and poles, as "zero" is less technical than "pole", and generally defined first. Also, the plural is justified by the fact that in complex analysis poles and zeros are generally considered as a whole. D.Lazard (talk) 14:55, 18 December 2017 (UTC)

My example failed to illustrate my point. Of course, in practice we can work around that situation by combining the two links. But I meant to illustrate the asymmetry in our article structure. After your proposal, we would have the following:

[[Zeros and poles]]
[[Zero of a function]]	no [[Pole of a function]]

That said, you make a good point that the zeros and poles (and, btw, I agree with the order) are generally considered as a whole. Maybe, then, we don't need a separate article [[Zero of a function]], either, and should all merge into [[Zeros and poles]]? — Sebastian 15:34, 18 December 2017 (UTC)

As far as I know, poles are mainly used in the context of complex analysis. This is certainly the case when zeros and poles are associated. Thus Zeros and poles refer implicitly to Zeros and poles (complex analysis), but the disambiguation is not really needed. On the other hand, "zero of a function" has a much wider usage (e.g. zeros of a polynomial, critical points as zeros of the gradient, ...). D.Lazard (talk) 17:13, 18 December 2017 (UTC)

Your hunch is right; poles are mainly used when they are associated with zeros. I searched Special:WhatLinksHere/Pole (complex analysis) to see if I could find links to Pole (complex analysis) where coverage of zeros would provide undue ballast to the reader, but only found a few borderline cases, such as

• Undefined (mathematics),
• Singularity (mathematics),
• Regular singular point,
• Elliptic function,
• Differential of the first kind.

By contrast, more than half of the investigated articles contain a link to Zero of a function, as well. A number of articles, such as the following, would even benefit from a link to a combined article:

• Pole–zero plot, an article which one should take a close look at before creating Zeros and poles.
• System analysis,
• Control theory,
• Root locus (currently contains no link to zero, but makes extensive use of the term),
• Nyquist plot,
• Logarithmic derivative,
• Comb filter,
• Nonlinear control.

Conversely, there are many articles among Special:WhatLinksHere/Zero of a function where coverage of poles would provide undue ballast. I therefore now support your suggested article partition and the suggestion that Pole (complex analysis) should redirect to Zeros and poles.

Only one suggestion of yours still doesn't make sense to me: Why should Zero (complex analysis) redirect to Zeros and poles, and not to Zero of a function? An investigation of Special:WhatLinksHere/Zero (complex analysis) appears to yield the same results as that of Special:WhatLinksHere/Pole (complex analysis): many articles containing links to both zeros and poles, and a number of articles would benefit from combining these links, such as

• Complex number,
• Complex plane,
• Valuation (algebra),
• Linear filter,
• Positive-real function,
• Argument principle,
• Analogue filter,
• Mathematics of radio engineering,
• Z-transform,
• Bode plot,
• Foster's reactance theorem.

Your suggested redirect might make sense if there were many instances of articles linking to Zero (complex analysis) without linking to Pole (complex analysis), but needing explanation of poles. I did not find any article where that would be the case. If we do fine one or two, the effort to correct the links will be negligible; no need to rely on a redirect. In any case, there are more articles that intentionally only link to Zero (complex analysis) without needing an explanation of poles. Examples:

• Stark conjectures (via redirect Order of vanishing),
• Zero (disambiguation)‎,
• Factor theorem,
• Weierstrass factorization theorem (poles only needed for extended theorem),
• Hurwitz's theorem (complex analysis),
• Unit hyperbola.

I also found some articles that contain a link to zero, and later a link to pole, such as

• Entire function,
• Doubly periodic function ("pole" is not actually linked, but it wouldn't hurt.)

In those cases, it probably would be better to link to Zero of a function first, and later to Zeros and poles (as "pole" via piped link or redirect). — Sebastian 01:03, 19 December 2017 (UTC)

Correct links in articles is a different question from names of articles and redirects. Whichever structure of articles partition is chosen, some links should be modified, at least for avoiding double redirects. In any case, Zero (complex analysis) should redirect somewhere, and Zero of a function is not a bad target. However, this target is somehow a WP:broad-concept article, considering different, but related concept, which are detailed in linked articles. These are zero of a univariate function (Root-finding algorithm), polynomial roots (Fundamental theorem of algebra and Properties of polynomial roots), and zero sets (infinite sets of zeros, Zero set was a separate article that I have just redirected here). A section about zeros of analytic functions (holomorphic functions) is presently missing. Even if this gap is filled, a reader interested in this aspect knows that this is a part of complex analysis, and should presumably prefer going directly to Zeros and poles. By WP:LEAST, it seems thus better to redirect Zero (complex analysis) to Zeros and poles. D.Lazard (talk) 09:39, 19 December 2017 (UTC)

I wholeheartedly agree with keeping WP:LEAST in mind, but we differ in what we think a reader is looking for. You base your opinion on an educated guess for which we have no evidence. The only evidence we have is what editors were looking for, for which we even have a relevant sample size. All it takes to apply it to our purpose is the assumption that the average editor is representative for the average reader, which I believe is reasonable. If one accepts that as evidence, it overwhelms any speculation. We also disagree on whether a WP:broad-concept article is a good redirect target. I think it often is, as can be seen by the fact that even our example article Triangle center is the target of redirects like Center function.

Hence, I am not convinced by your conclusion. But I admit that Zeros and poles, as you envision it, will not be a bad target, either. It doesn't matter much which of the two articles we choose as a link target, since it will be easy to find one from the other. Let that not distract us from the major task at hand, the merger of the articles, on which we agree. — Sebastian 11:34, 19 December 2017 (UTC)

Moved Pole (complex analysis) to Zeros and poles and merged Zero (complex analysis) into it. D.Lazard (talk) 21:22, 14 January 2018 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Simple roots and/or multiplicity, for several variables

I was looking for a definition of "simple root" for a function in more than one variable. I did not find such. Somewhere else, I found a definition of multiplicity of a root w "for a meromorphic function, if f(z)/(z-w)^n ..." but this assumes that we have one complex variable, while I'd be interested in the case of several real variables. In particular, I'd like to know whether it is acceptable to say that "simple" means (roughly) f' ≠ 0. — MFH:Talk 23:21, 21 May 2018 (UTC)

I think "simple root" and "multiplicity", together with fixed coordinates, somewhat lose their importance in the context of more than one variable. Maybe "regularity", "full rank", or "rank deficit" of differential maps are along your interests. Purgy (talk) 06:27, 22 May 2018 (UTC)