Talk:Discriminant

message: 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)
replies: text: 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)
thread_title:  [[Matrix_representation_of_conic_sections]] 

message: 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)
replies: text: 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)
thread_title:  cubic and other degrees 

message: In this section it 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)
replies: text: 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)
thread_title:  Possible error in Section "Quadratic Forms" 

message: 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)
thread_title:  Merger proposal