Talk:Multiplicity (mathematics)

Multiplicity of a root of a polynomial

Hey guys, I appreciate the precise mathematical definition of the multiplicity of the zeros of a function, but how about a simple one line explanation that the multiplicity of a root is the number of times that root is repeated? Followed by a simple example, like the root of f(x) = (x-1)^3 is 1, with a multiplicity of 3, because it occurs 3 times. —Preceding unsigned comment added by Nedunuri (talk • contribs) 18:59, 20 Oct 2006 (UTC)

I second that. Myers6609 (talk) 01:15, 7 January 2009 (UTC)[reply]

I added- Multiplicity can be thought of as "How many times does the solution appear in the original equation?". hopefully making it explicit and more clear.Nickalh50 (talk) 19:09, 2 May 2011 (UTC)[reply]

The problem is that this is just informal language; there is no such thing as "occurring as a solution" in general; a value either is a solution of a problem, or it isn't. The reason that one can define multiplicity in the case of a polynomial equation is that there is a method to "remove" a solution (namely division of the polynomial by a factor corresponding to the root), and that only if the root still persists as a solution of te new problem does one consider it to occur more than once. The text already says this, and the lead section already says that multiplicity in general means being present multiple times, when a precise meaning can be given to that. So what does the sentence really add? Marc van Leeuwen (talk) 03:39, 3 May 2011 (UTC)[reply]

Root / zeros

The article should be consistent in using root or zeros. Root is the more common in scientific use. Thomas Nygreen (talk) 22:52, 5 December 2007 (UTC)[reply]

Then edit it... However, in analysis, I think that the zeroes of a function is more common than roots. mattbuck (talk) 23:02, 5 December 2007 (UTC)[reply]

In my experience, root, zero, & solution are used more or less interchangeably in this context. We need links to Root_of_a_function and I added a few.Nickalh50 (talk) 20:42, 2 May 2011 (UTC)[reply]

Multiplicity of a zero section needs an added explanation of why it won't overestimate the multiplicity

This could be something simple, like "if k is an underestimate, then the limit evaluates to 0, and if k is an overestimate, then the limit evaluates to ∞, which is not a real number." Or it could give a full-fledged derivation for why both of these is the case. Regardless, an explanation is needed, IMHO.

It could also be pointed out that k need not necessarily be an integer. Take f(X) = X^1.5 at 0, for a simple example.

I don't really have any real expertise on this topic; I'm just going by what I see on face value (plus a little, I guess), so if there's any mistaken impression here, feel free to correct me, and there will be no hard feelings. :) --69.91.95.139 (talk) 01:00, 15 April 2009 (UTC)[reply]

"None"?...

* A little knowledge is a dangerous thing.

- A little Learning is a dangerous Thing;
  Drink deep, or taste not the Pierian Spring:
  There shallow Draughts intoxicate the Brain,
  And drinking largely sobers us again. ~ Alexander Pope
  --Jerzy•t 07:38, 13 December 2018 (UTC)[reply]

Multiplicity of a zero section should be rewritten for precision or removed

In its current state, this section is not correctly formulated, and is a very bad illustration of the general notion of multiplicity. Since nothing is required of the function (an indeed the interval is not even required to not be reduced to a point), it need not be continuous at its zero; if it isn't, I think none of the definitions given applies, so one can only conclude that the zero has no multiplicity at all (not even multiplicity zero). Other problematic points are infinite multiplicities (which are not allowed in the most common definition of multiset; even those who prefer to be more liberal make it a cardinal number, not just infinity) and multiplicities that are in other ways undefined (what is the multiplicity of the zero of $x\mapsto x^{\alpha }$ for any positive real number α?). I insist that while infinite multiplicities can be admitted in some cases (any "root" of the zero polynomial), ordinary use of "multiplicity" does not allow it to be just any real number (and why not quaternions for tomorrow's new application?) Marc van Leeuwen (talk) 08:51, 3 April 2011 (UTC)[reply]

Root

Just a suggestion, for easier reading and to be a bit more concise, to change the first 2 sentences of the section Multiplicity of a root of a polynomial,

from

Let F be a field and p(x) be a polynomial in one variable and coefficients in F. An element a ∈ F is a root of multiplicity k of p(x) if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)^ks(x).

to

Let p(x) be a polynomial in one variable. A solution x = a of p(x) = 0 is a root of multiplicity k if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)^ks(x).

--Bob K31416 (talk) 16:09, 4 November 2017 (UTC)[reply]

In the standard mathematical terminology, a root of the polynomial p(x) is, by definition, a solution of the equation p(x) = 0. Thus the formulation "a solution of ... is a root of multiplicity ..." may be confusing for some readers, as using two different words for the same concept. For this reason, I suggest the following formulation:

Let p(x) be a polynomial in one variable. A root a of p(x) (that is a solution of the equation p(x) = 0) has the multiplicity k if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)^ks(x). One says also that a is a root of multiplicity k.

If you agree, feel free to insert this in the article.

By the way, if a is not a root, one may say that it is a root of multiplicity zero. This may be useful for simplifying some statements and proofs. IMO, it is not useful to mention this here. D.Lazard (talk) 16:43, 4 November 2017 (UTC)[reply]

Other senses: usage by Cantor not (evidently) the common-language meaning.

@D.Lazard: reverted an edit of mine to explain the usage of "multiplicity" by Cantor at Absolute infinite, saying that this is the common-language meaning. It seems to me however that neither a set, nor an ordered set (which he seems to have meant here), is covered by the definitions at wikt:multiplicity. I had added a section Other Senses to this page to explain that usage, as linking "multiplicity" to cardinal number seems to me to help no-one. I changed that to link to my new section, but since that has been removed, it just links to the entire page, which is even worse. What I added was this:

Some older or translated texts use this term differently:

Georg Cantor, as translated into English, appears to use it to mean an ordered set (text in square brackets not present in original):^[3]^[1]

A multiplicity is called well-ordered if it fulfills the condition that every sub-multiplicity has a first element; such a multiplicity I call for short a "sequence".

...

Now I envisage the system of all [ordinal] numbers and denote it Ω.

...

Therefore:

The system Ω of all [ordinal] numbers is an inconsistent, absolutely infinite multiplicity.

References

^ ^a ^b Georg Cantor (1932). Ernst Zermelo (ed.). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Berlin: Verlag von Julius Springer. Cited as Cantor 1883b by Jané; with biography by Adolf Fraenkel; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, ISBN 3-540-09849-6.
^ The Rediscovery of the Cantor-Dedekind Correspondence, I. Grattan-Guinness, Jahresbericht der Deutschen Mathematiker-Vereinigung 76 (1974/75), pp. 104–139, at p. 126 ff.
^ Gesammelte Abhandlungen,^[1] Georg Cantor, ed. Ernst Zermelo, Hildesheim: Georg Olms Verlagsbuchhandlung, 1962, pp. 443–447; translated into English in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jean van Heijenoort, Cambridge, Massachusetts: Harvard University Press, 1967, pp. 113–117. These references both purport to be a letter from Cantor to Dedekind, dated July 28, 1899. However, as Ivor Grattan-Guinness has discovered,^[2] this is in fact an amalgamation by Cantor's editor, Ernst Zermelo, of two letters from Cantor to Dedekind, the first dated July 28 and the second dated August 3.

Can anyone suggest a better way of resolving this confusion? I do not now have time to look into this further, but perhaps we have somewhere else an article which includes a description of his ideas of "multiplicity", and we could link Absolute infinite there, and maybe add a hat note here.

PJTraill (talk) 12:06, 6 November 2022 (UTC)[reply]

I have added the template {{talkref}} in the previous post, and removed a duplicate signature.

It is clear that Cantor or its translator used "multiplicity" for what is presently called a set with several elements. This the common meaning of the term as expressed in Wikt:multiplicity ("A large indeterminate number"). Per WP:NOTDICT, the meanings of "multiplicity" that refer to other concepts than the one of this article do not belong to it. So, Wikipedia rules justify my revert. The only thing that you can do is to use {{wikt}}. In any case, the details of Cantor's terminology have no encyclopedic value. D.Lazard (talk) 13:03, 6 November 2022 (UTC)[reply]