Talk:Discriminant

Matrix_representation_of_conic_sections
A recent edit removed mention of Matrix_representation_of_conic_sections. It seems to me that this isn't so well known, and deserves mentioning along with the rest of the part on conic sections. How it should be mentioned, I don't know. Gah4 (talk) 19:03, 2 August 2020 (UTC)
 * Matrix_representation_of_conic_sections is not specifically related to the discriminant and does not mention the discriminant. So there is no reasons for linking to it in Discriminant. There are links to Matrix_representation_of_conic_sections in Conic section and Quadratic form, which are the two main article where links to Matrix_representation_of_conic_sections are really important. Also, it is linked to in all the articles containing the template template:Matrix classes. So, there is a sufficient coverage of this article in Wikipedia, and it is not useful to link to it in articles where this is not fully relevant. D.Lazard (talk) 19:24, 2 August 2020 (UTC)

cubic and other degrees
The first paragraph mentions the quadratic equation and its discriminant (good). Then it goes on to mention cubic equations and others, without mentioning that their discriminant is different. That isn't so obvious as it might seem. Can we make it more obvious that the are different, without giving their form (yet)? Gah4 (talk) 19:07, 2 August 2020 (UTC)
 * The second sentence of the article is "The discriminant of a polynomial is generally defined in terms of a polynomial function of its coefficients". This clearly establishes that polynomials with different coeefficients have (in general) different discriminants. This clearly applies to discriminants of quadratic and cubic equations. Moreover, the lead is here for giving a summary, and for the details, the reader who finds a sentence of the lead confusing has to read the body of the article. Here we have separate sections for the qudratic and the cubic cases. D.Lazard (talk) 19:35, 2 August 2020 (UTC)

There are no pages for quadratic polynomial or cubic polynomial
There are no pages for quadratic polynomial and cubic polynomial because they are redirects. There are, however, pages for quadratic function and cubic function and there are pages for quadratic equation and cubic equation. Regardless of what the "correct" term is, if you actually read the content of those articles, it is very clear that one set is more relevant to a discussion about the discriminant of a polynomial than the other. I tried making the very obvious link correction along with a very detailed explanation but it was swiftly reverted by someone who left a note indicating that they had not done their due diligence.

Might seem like a small thing but I'm actually studying this topic at the moment. And quadratic function and cubic function were unhelpful enough for me track down the correct pages so I figured I'd save the next person some time. Robo042 (talk) 00:57, 13 July 2021 (UTC)
 * Which page do yo need to track for learning about discriminants? The page Discriminant is there for that. If some information is lacking, then it must be added to this page. However Wikipedia is not a textbook; so for learning a subject, it is often better to work on a textbook.
 * Apparently, you have not understood that the wikilinks in the text of the article are not here to signal related articles, but for providing a definition of the technical terms that are used (here quadratic polynomial and cubic polynomial). This is a big difference between Wikipedia and a textbook, because, in a textbook, the author knows which terms are supposed to be known by the reader, and must define the others. Here, it is true that the theory of discriminants and the theory of polynomial equations are strongly related, and that, in degree 2 and 3, all properties of a polynomial equation result from its discriminant. Therefore you are right that quadratic equation and cubic equation must be linked. It is the reason for which I have linked these pages, and more specifically their sections "Discriminant" at the top of the respective sections of this article, by mean of a see also template. D.Lazard (talk) 12:21, 13 July 2021 (UTC)


 * I'm reading to supplement my math studies. Something I would highly recommend to folks looking to improve their English skills as well.
 * I'm fine with your solution. And I would have been fine with your reverts if it wasn't very obvious that you had performed them before fully reading the detailed edit descriptions.  In the future, please be sure to do your due diligence before reverting logical and well-explained edits.
 * Robo042 (talk) 12:38, 13 July 2021 (UTC)

Possible error in Section "Quadratic Forms"
In this section it [https://en.wikipedia.org/w/index.php?title=Discriminant&oldid=1142822728#Quadratic_forms:~:text=Over%20the%20reals%2C%20if%20the%20discriminant%20is%20positive%2C%20then%20the%20surface%20either%20has%20no%20real%20point%20or%20has%20everywhere%20a%20negative%20Gaussian%20curvature.%20If%20the%20discriminant%20is%20negative%2C%20the%20surface%20has%20real%20points%2C%20and%20has%20a%20negative%20Gaussian%20curvature. is stated] that "Over the reals, if the discriminant is positive, then the surface either has no real point or has everywhere a negative Gaussian curvature. If the discriminant is negative, the surface has real points, and has a negative Gaussian curvature".

However, the correct statement shouldn't be "Over the reals, if the discriminant is positive, then the surface either has no real point or has everywhere a positive Gaussian curvature. If the discriminant is negative, the surface has real points, and has a negative Gaussian curvature"? For instance, taking A as the identity matrix we have the equation of a sphere, whose Gaussian curvature is positive everywhere. Saung Tadashi (talk) 14:11, 10 March 2023 (UTC)
 * The formulation is correct, although it should be made clearer that the discriminant refers to the equation of the quadric in the projective space (homogeneous equation), and the Gaussian curvature refers to a non-homogeneous equation in 3D space. In the case of a unit sphere, the homogeneous equation is $$x^2+y^2+z^2-t^2=0,$$ and the discriminant is −1, as the determinant is of a 4×4 diagonal matrix that contains three 1 and one −1. D.Lazard (talk) 14:44, 10 March 2023 (UTC)

Merger proposal
I propose merging Fundamental discriminant into Discriminant. In fact, almost all the content of Fundamental discriminant is more detailed in Discriminant. The only exception is the list of discriminants of quadratic number fields, that is easier to find in Quadratic number field or in Discriminant of an algebraic number field, the latter being linked to in the eponym section of Discriminant. Moreover, “fundamental discriminant” is generally simply called “discriminant” in modern textbooks, and this makes difficult to find this article. D.Lazard (talk) 13:30, 3 April 2023 (UTC)