Talk:Discriminant/Archive 1

Archive 1

Discriminant of a polynomial

I have found that the TeX code does not compile if the matrix has more than 10 colons. Maybe there is a way around this? The following matrix seems, to me, sufficient for comprehension:

${\begin{pmatrix}1&a_{n-1}&a_{n-2}&.&a_{0}&0&.&.&.&0\\0&1&a_{n-1}&.&.&a_{0}&0&.&.&0\\0&0&1&.&.&.&a_{0}&0&.&0\\.&.&.&.&.&.&.&.&.&.\\.&.&.&.&.&.&.&.&.&.\\0&0&...&1&a_{n-1}&a_{n-2}&.&.&.&a_{0}\\n&(n-1)a_{n-1}&(n-2)a_{n-2}&...&0&0&.&.&.&0\\0&n&(n-1)a_{n-1}&...&1a_{1}&0&0&.&.&0\\.&.&.&.&.&.&.&.&.&.\\.&.&.&.&.&.&.&.&.&.\\0&0&...&n&(n-1)a_{n-1}&(n-2)a_{n-2}&.&.&1a_{1}&0\\0&0&...&0&n&(n-1)a_{n-1}&(n-2)a_{n-2}&.&.&1a_{1}\\\end{pmatrix}}$

and note that there should be an (n-2) factor to the a_{n-2} terms in the two last rows. If there is no opposition, I think that this should replace the current "array". Gene.arboit 03:17, 12 August 2005 (UTC)

I combined some dots into \ldots, and compressed it into less than 10 columns, see if the matrix is comprehensible. Wang ty87916 15:55, 2 September 2006 (UTC)

Discriminant of an algebraic number field

The stuff on the discriminant of an algebraic nubmer field should be a separate article. Although there are connections between discriminants of polynomials and of number fields, they are really two quite separate topics. Dmharvey File:User dmharvey sig.png Talk 12:54, 29 August 2005 (UTC)

There may be a reason to split that out; but the 'separateness' is quite debatable. Charles Matthews 14:25, 29 August 2005 (UTC)

Latex Help

Help! My LaTex isn't working! I'm trying to LaTex the major formulas,but it says: Failed to Parse on some of them.

It appears that you somehow inserted some invisible unicode characters. Maybe you accidentally hit some weird key combination. In any case, I have reverted the changes, according to the guidelines at Wikipedia:Manual_of_Style_(mathematics). Dmharvey 08:08, 1 January 2006 (UTC)

Extra Topic Needed

Anyone know about Discriminant Functions in Statistics?

Please add, if you do.

Disambiguation

As there are many types of discriminants in mathematics: polynomial discriminant, elliptic discriminant, modular discriminant, fundamental discriminant, conic section discriminant, metric discriminant, etc. see MathWorld. So can we move the main portion of this article to Discriminant (polynomial) and create separate articles for all other discriminants, and set up an disambigution page?

I agree - also in discriminants (polynomial) we ought to be (a)distinguishing between algebraic (D) and geometric(\Delta) discriminants (functionally equvalent, but conceptually distinct - see paper in links) and (b) we should be a bit more purist regarding the exact definition of these algebraic discriminants: to I would prefer us to use something like D_n = (\alpha_1 - \alpha_2)^2 (...)^2 (...... etc to define the basic algebraic discriminant (of degree n), and subsequently show that other discriminants of the algebraic variety are either factors or multiples of this, and hence remove the confusion commonly experienced by many students wrt this topic. Halothane (talk) 17:42, 8 March 2009 (UTC)

Confusing article tag

I have to agree that this article is very confusing as it stands. If some high-school freshman wanders in here trying to read about the discriminant his algebra teacher told him about, the 10x10 matrix will probably give him a case of the fantods, and drive him away from dry-as-dust mathematics for the rest of his life!

I think the headline article discriminant should present a simple explanation for maybe the quadratic and cubic polynomials, and move the results from complex analysis / higher theory of polynomial equations into a separate section at least, or maybe even a separate article. I'll give it a go, but thought I'd ask for feedback first.  ;^> DavidCBryant 14:18, 13 December 2006 (UTC)

Well, I went ahead and rewrote the article. It can still use lots of improvement, I'm sure. I added a section having some formulas (the ones that one always needs to look up). I put the quadratic formula stuff in a subsection. Now the general definition is further down, and that may make the article less offputting. I also fixed the definition (I think). Previous versions claimed the discriminant is equal to the resultant of p and p'. That is not true. --345Kai 10:17, 14 December 2006 (UTC)

This is a problem with pretty much every math article on Wikipedia. 61.175.120.34 (talk) 01:43, 20 October 2017 (UTC)

Error on the first sentence?

I didn't want to edit without checking with the community first, but in the first sentence it says the discriminant is zero if and only if the polynomial *does* have multiple roots in the complex numbers, which is the opposite of the right answer, right? Omgoleus 14:23, 8 February 2007 (UTC)

• That statement seems to be correct; see the the section on the quadratic formula. Akriasas 16:49, 12 March 2007 (UTC)
• Either way, the article contains a contradiction. The first sentence states that the discriminant of a polynomial is zero when the polynomial has multiple complex roots, whereas the section titled "The discriminant of the quadratic formula" says that a quadratic equation whose discriminant is zero will have two REAL roots. Which part is correct? Frito (talk) 00:58, 12 December 2007 (UTC)
  • It's not in contradiction because the reals are a subset of the complex numbers. I don't know how to rewrite the sections to make that less confusing. Akriasas (talk) 20:57, 12 December 2007 (UTC)

The whole paragraph is wrong

The following paragraph is not easy to fix, but does not belong to the lead anyway, in my opinion:

The concept of discriminant has been generalized to other algebraic structures besides polynomials, including conic sections, quadratic forms, and algebraic number fields. Discriminants in algebraic number theory are closely related, and contain information about ramification. In fact, the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications.

• Conic sections are not algebraic structures, while their discriminants are discriminants of polynomials.
• Quadratic forms are polynomials. However, they are polynomials of several variables. Is this what caused the confusion?
• Discriminants of algebraic number fields are, of course, part of algebraic number theory.
• The sentence about ramification is extremely obscure.

I propose to delete the whole paragraph. Arcfrk 16:20, 22 March 2007 (UTC)

Matrix wrong?

The matrix shown in the page is billed as (2n - 1)×(2n - 1) but beginning with the row n from the end (the p' rows), the rows start as:

a_n a_{n - 1} … a₁ 0 … 0

and end as

0 … 0 a_n a_{n - 1} … a₁

with the first a_n in the last row lined up with the first 0 on the first line. This would imply n a_i entries and n zeroes in each row. making a 2n×2n matrix. It doesn't agree with the example, either.

I don't know enough math to be able to say whether the matrix is wrong or the dimensions/example (although I suspect the matrix), but perhaps someone should look into this.

Interesting article, though

139.85.238.27 (talk) 18:44, 17 April 2008 (UTC)Doug Dean

Thanks Doug; the matrix is correct – you just counted an extra 0: there are actually n–1 zeros in these rows: since a₁ lines up with a_n, the zeros thus line up with a_n–1 through a₁, and thus there are n–1 of them.

—Nils von Barth (nbarth) (talk) 14:37, 28 April 2009 (UTC)

Formula section

Two questions: 1. shouldn't the section be called formulae (plural)?
2. should we really include formulae for the monic form of the polynomials when they obviously and trivially follow from the more general formula? --Cynic (talk) 00:06, 8 December 2008 (UTC)

About cubic Roots

In this article, at Paragraph "Nature of the roots" and concerning the Cubic polinomial, it states

• Δ > 0: the equation has 3 distinct real roots;
• Δ < 0, the equation has 1 real root and 2 complex conjugate roots;
• Δ = 0: at least 2 roots coincide, and they are all real.

While I found several documents where it is explained it this way:

• Δ > 0: one root is real and two are complex conjugates;
• Δ = 0: all roots are real and at least two are equal;
• Δ < 0: all roots are real and unequal.

Which is basically the opposite of what is explained in the article. Here the links to two places where I found above explanations: here (right at 'Step 3') and here (after equation [69] )

It seems to me really misleading and I don't know who's right, but one would rather trust better the others, since more examples and explanations uses that logic. — Preceding unsigned comment added by Faabiioo (talk • contribs) 17:06, 12 March 2011 (UTC)

Name

Has anyone figured out why it is called a "discriminant"? If anyone can figure that out, they should post it in this article. NO SPECULATION.. although i suppose you knew that already.69.153.0.86 (talk) 21:22, 16 April 2012 (UTC)

Discriminant from the root 'discriminate' because the discriminant separates different possible regimes for the roots of a polynomial - i.e. it discriminates them. Tweet7 (talk) 16:25, 4 February 2013 (UTC)

I may add that it is the French word for "discriminating", like "determinant" is the French word for "determining". D.Lazard (talk) 18:36, 4 February 2013 (UTC)

Wrong description in Homogeneity section

If you compute the terms for degree 4 by the partition example given at the end of the Homogeneity section, then this would give you also 12 = 2 + 2 + 2 + 2 + 2 + 2, which corresponds to the term c^6, which doesn't appear in the discriminant (as well as some other terms that are 'predicted'). Therefore, the partition approach is (sadly) wrong. Can someone remove/edit this? (I have no idea how to do it. I am happy if this comment will appear where I want it to be...) — Preceding unsigned comment added by 129.67.186.112 (talk) 15:04, 3 May 2013 (UTC)

Edit by me: OK, the coefficients can be zero...I still vote for a slight edit to make that clear... — Preceding unsigned comment added by 129.67.186.112 (talk) 15:07, 3 May 2013 (UTC)

I have edited the section (and also quasi-homogeneous polynomial) to clarify. I have left the interpretation in term of partitions (which is not essential) for the example of the third degree. Some more edits are needed but I think that now it is no more confusing. (Also, please, sign your posts with four tildes ~~~~). D.Lazard (talk) 21:24, 3 May 2013 (UTC)

Still contradictions and confusion

Very unhelpful article. I, too, was immediately struck by the contradiction between the very beginning and a more general sentence just a little further down, as has been noted by another reader. The contradictions between this article and others (also mentioned) have still not been exlained and/or fixed. I came here to understand the concept, and now understand less than I did before! — Preceding unsigned comment added by 82.68.94.86 (talk) 19:50, 26 October 2013 (UTC)

Please be more explicit. What is the contradiction and which are the implied sentences? D.Lazard (talk) 22:21, 26 October 2013 (UTC)

The phrase "Failed to parse" occurs twice in the current version. — Preceding unsigned comment added by 86.181.9.123 (talk) 11:53, 7 February 2014 (UTC)

The origin of the name

Hi, Does anyone know the origin of the term Discriminant and if it has any connection to discrimination? 109.64.50.70 (talk) 06:41, 2 May 2014 (UTC)

"Discriminant" is the French word for "discriminating" of "distinguishing". Its use in mathematics is much older than the meaning of "prejudicial treatment" for discrimination. D.Lazard (talk) 09:24, 2 May 2014 (UTC)

Great, thank you very much. 109.64.50.70 (talk) 13:24, 7 May 2014 (UTC)

Figures of the discriminant variety

Salix alba‎ has recently added two figures in section "Formulas for low degrees". This certainly improves the article from an aesthetic point of view, but, in my opinion not from an encyclopedic point of view.

The figure for the cubic case seems erroneous: the dark-green sheet on the right, behind the curve of cusps seems to be an artifact of the used software. In fact, the intersection of the surface with any plane b = constant is a smooth deformation of the cuspidal cubic obtained for b = 0. This deformation is the inverse of the transformation from the general cubic equation to the depressed cubic. Thus the surface is topologically and algebraically the product of a cuspidal cubic by a smooth curve. This should be clear by looking to the figure, and it is not, because of the dark-green sheet. Also, the surface is invariant by changing the signs of b and d (typo in the caption), and this is not clear by looking on the figure.

The problem with the quartic case is different. Although I know rather well this surface, I am unable to understand the figure. The figure has a parabola of self crossing points, and a skew cuspidal cubic of cusps. Half of the parabola is isolated in the sense that near these points, the surface is (from a real point of view) reduced to a curve. The projection of the cubic curve on the plane e = 0 is an ordinary cuspidal cubic; its projection on the plane d = 0 is twice a parabola. Both singular curves intersect at b = c = d = 0. I am unable to recognize any of these properties on the figure. IMO, this figure would better replaced by the drawing of three plane curves corresponding to the sections of the surface by the planes c = -1, c = 0, c = 1: one obtains the other plane sections by remarking that the surface is invariant by the transformation c → u²c, d → u³d, e → u⁴e. Thus, the shape of the curve does not change when c varies, keeping a constant sign. For c > 0, the plane section looks like a parabola (it is not) with an isolated singular point inside (corresponding to two double complex conjugate roots); otherwise, the interior of the curve correspond to four non real roots, and the exterior to two real and two non real roots. For c = 0, the curve is also parabola like, with the same characterization of its interior and its exterior. For c < 0, the curve has two cusps and a crossing point. It is symmetric with respect to d = 0. Between the three singular points, the curve is a curvilinear triangle inside which the polynomial has four real roots. Starting from the crossing point there are two parabolic branches, inside which the polynomial has four non real roots. Outside these two regions, there are two real and two non-real roots.

Again, for being useful, a figure must clarify these properties, and I do not know a better way that the above three plane sections. Unfortunately, I am unable to produce these drawings myself.

This may look as original research, but it is not. The classification of Quartic function#Nature of the roots is based on the study of this discriminant surface, as this appears clearly form the title of the source of this section. D.Lazard (talk) 13:39, 6 April 2015 (UTC)

Thinking again on the subject, it appears that one could have decent 3D plots by using rational parameterization of the surface. For the quartic discriminant such a parameterization is

c = c

d = 4t³c³ + 2tc²

e = 3t⁴c⁴ + t²c³

(As, for c fixed, we have a quartic with three singular points, this parameterization has been obtained by the standard way: the point of the curve is the last intersection point with a conic passing through the 3 singular points and the point d = e =0.)

A rational parameterization of the cubic discriminant surface is

b = b

c = ⁠b² – t²/3⁠

d = ⁠b³ – 3bt² + t³/27⁠

Both parameterizations have two important advantages: firstly, parametric plotters are commonly much better than implicit plotters; secondly, as the first variable is also a parameter, it is easy to highlight the curves on the surface where this variable is constant. D.Lazard (talk) 10:10, 7 April 2015 (UTC)

For the cubic, you could see the top surface of the back sheet, this was not an artefact more an indication of quite how dramatically the cusp swings for large b. I've redone the picture with a lower range for b, showing the cusp clearer and with no backside visable.

The swallowtail surface is rather well known in singularity theory and Catastrophe_theory and it arrises as the discriminant, it also featured in Dali's last work The Swallow's Tail. I've followed Bruce, J. W.; Giblin, P. J. (1984), Curves and Singularities, p128/129, for this image. You can verify the surface by taking slices

• Plot with x=c, y=d, e constant
• Plot with x=c, y=e, d constant
• Plot with x=d, y=e, c constant

you will need to switch to the parameter tab and change the value of the constant to see. --Salix alba (talk): 14:02, 7 April 2015 (UTC)

Above parameterizations of the surface of the zeros of the discriminant are not the best one. In fact, the hypersurface of the zeros of the discriminant of a polynomial of any degree is easily parameterizable by using the following remark: The discriminant is zero if and only if the polynomial and its derivative have a common root (which is a double root). This gives two equations that are linear in the linear coefficient and the constant coefficient of the polynomial. Solving this system of equations in terms of these two coefficients is thus easy. This gives the following parameterizations (with above notation). For the quartic:

c = c

d = –4x³ – 2cx

e = 3x⁴ + cx²

(one passes from one parameterization to the other by setting x = –ct)

For the cubic:

b = b

c = –3x² – 2bx

d = 2x³ + bx²

In both case, x is the value of the double root. D.Lazard (talk) 16:18, 10 April 2015 (UTC)

Should we emphasize that the leading coefficient is not zero.

Lklundin has made this edit and has restored it after my revert. I disagree with this edit for several reasons. One is the summary of my revert: No, a is an indeterminate, and the discriminant, being a polynomial in the coefficients, remains defined for polynomials of lower degree than the degree used for computing the discriminant. Because of the lack of place in edit summaries, this is incomplete and rather elliptic. Thus the reasons of my revert require to be detailed.

1. A formula such as $ax^{2}+bx+c,\quad a\neq 0$ is formally incorrect, the comma being not a conditional operator.
2. The second part of the this formula is redundant, as it has just been said that one considers a quadratic polynomial
3. Discriminants are traditionally defined for polynomials with indeterminate coefficients. If a is an indeterminate, comparing it to zero is a nonsense.
4. Discriminants are frequently used for polynomials whose coefficients are polynomials in other variables. In this case, the universal property of the discriminant asserts that substituting some values to the variables appearing in the discriminant, gives the discriminant of result of the same substitution in the original polynomial. This remains true if the degree of the polynomial decreases under this substitution, with the convention that the discriminant after substitution is computed as if the degree would be the same. This convention is useful, for example, in the study of plane algebraic curves defined by an implicit equation: The zeros of the discriminant with respect to y are exactly the values a of x for which the line x – a = 0 has a multiple contact with the curve, a tangent in general, or an asymptote (contact at infinity) if a is a root of the leading coefficient in y. Thus the discriminant may provide useful information, even if the leading coefficient is zero.

For all these reasons I will restore again the old version of the article. Please do not modify it again without getting a consensus on this talk page. D.Lazard (talk) 15:16, 22 July 2015 (UTC)

The definition

The discriminant should be introduced to the reader as being:

$\prod _{i<j}{\left(r_{i}-r_{j}\right)}^{2}$

I say this for a few reasons. Firstly it's easier to understand for a beginner (the people this article should be aimed at) as it doesn't require knowledge of resultants or Sylvester matrices. The important properties of the discriminant are also immediately clear in this setting. Additionally the polynomial $\prod _{i<j}{\left(r_{i}-r_{j}\right)}^{2}$ is of Galois theoretic significance. For a general polynomial of degree 5 or greater, the square root of this value ($\prod _{i<j}{\left(r_{i}-r_{j}\right)}$) is as far as you can get in solving the polynomial with radicals, since it is fixed by $A_{n}$ but not $S_{n}$. In fact this polynomial can be used to show that $A_{n}$ is a subgroup of $S_{n}$ as is the case in the article on parity. If the current convention in research applications is to define the discriminant as ${a_{n}}^{2n-2}\prod _{i<j}{\left(r_{i}-r_{j}\right)^{2}}$ that is unfortunate.

Also a sketch of a proof that the discriminant is the same thing as $\operatorname {Res} _{x}(A,A')$ would be very helpful. But I have no idea what the proof looks like, so it might not be possible. — Preceding unsigned comment added by 132.185.161.122 (talk) 12:19, 28 June 2017 (UTC)

I disagree that the discriminant should be first introduced as a product of differences of roots for the following reasons. However, I agree that this expression of the resultant is very important and must appear very soon in the article, which is presently the case. The reasons are:

• In the case of non-monic polynomials, the power of the leading coefficient which appears in the expression of the resultant is difficult to explain
• The formula in terms of differences of roots implies to know the existence of an algebraic closure or, at least, of a splitting field, which are non-elementary concepts. In the case of real or complex coefficients, the fundamental theorem of algebra suffices, but there are many applications of the discriminant other that number theory in characteristic 0.
• The formula in terms of differences of roots may be confusing when the coefficients are polynomials in other variables (important case, as the roots of the discriminant are the critical values of the projection on an hyperplane of an algebraic hypersurface)
• Many fundamental properties of the discriminant are much easier to deduce from the expression in term of a determinant than from the expression in terms of roots. Such are: the discriminant belongs to the ring generated by the coefficients of the polynomial; the discriminant behaves well under ring homomorphisms (including modular computation); homogeneity with respect to the coefficients; in the case of polynomials with numerical coefficients, Hadamard's inequality allows bounding the absolute value of the discriminant.
• For polynomials over a commutative ring that is not an integral domain, the formula in terms of differences of roots is meaningless.

I do not well understand your comment about $\textstyle \prod _{i<j}{\left(r_{i}-r_{j}\right)}:$ anything that can be said about this product may be said about the square root of the discriminant, which is $\textstyle a_{n}^{n-1}\prod _{i<j}{\left(r_{i}-r_{j}\right)}.$ Also, this is this latter product that appears in the usual quadratic formula.

The equivalence between the two definitions of the discriminant may be obtained by applying recursively the product formula of the discriminant (section "Product of polynomials"). This product formula results easily from the product formula for resultants: Res(AB, C) = Res(A, C) Res(B, C). This formula results itself from the fact that a resultant is a linear combination of the input polynomials, and is the smallest such constant linear combination if the coefficients are generic (that, is are indeterminates). Thus

$\operatorname {Res} (AB,C)=ABD+CE,$

for some polynomials D and E. Thus, this resultant is a linear conbination of A and C and also of B and C. Thus both Res(A, C) and Res(B, C) divide Res(AB, C), and the equality may be deduced from the degrees in the coefficients. This proves the formula in the case of generic coefficients, and the good behaviour of the resultant under ring homomorphisms implies that the product formula is always true. D.Lazard (talk) 17:30, 28 June 2017 (UTC)

Very bad beginning

The article begins by saying

"In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them."

No, no, no, no, no. This is the article called Discriminant, so you do not begin the article with describing a particular case of it without even saying what a discriminant is.

Is it not obvious that any math article on a particular topic must begin by saying what it is? That is certainly obvious to me. Because without that, people new to the subject will not know what it is talking about. Without any kind of definition at all first, a newcomer will find that other statements make no sense.2600:1700:E1C0:F340:6838:D7A6:BBFB:E78D (talk) 01:00, 23 August 2019 (UTC)

The quoted sentence says what the discriminant is. It is not a true definition because such a definition is too technical and too long for a first paragraph. However I agree that this first sentence is too technical for many readers, and particularly those who are interested only in the quadratic case. Therefore, I have put first the particular case (quadratic case), and moved the general description of what is a discriminant after it. I guess that this is the kind of modification that you are asking for. In this case, what is obvious for you is wrong here, as you will probably prefer the formulation that begin with a particular example before saying what it is. D.Lazard (talk) 08:07, 23 August 2019 (UTC)

The beginning to this article is poor. "In mathematics, the discriminant of the quadratic polynomial..." Wikipedia articles should not start with arbitrary specific examples. Why is the definition and general case delayed until after a discussion of a quadratic polynomial? I didn't navigate to this page to read about quadratic polynomials, I came here to learn the definition, meaning, and history of the discriminant. — Preceding unsigned comment added by EntangledLoops (talk • contribs) 01:44, 5 April 2020 (UTC)

It is convenient to start from the basic examples, as many people have heard of the quadratic case and ignore that there is a generalization to higher degrees. Nevertheless, I have added a sentence explaining that there is a general definition. D.Lazard (talk) 11:33, 5 April 2020 (UTC)