{"id":2522,"date":"2016-06-01T01:00:38","date_gmt":"2016-06-01T01:00:38","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=2522"},"modified":"2016-06-06T06:14:13","modified_gmt":"2016-06-06T06:14:13","slug":"discriminant-of-restricted-quintic","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2016\/06\/01\/discriminant-of-restricted-quintic\/","title":{"rendered":"Discriminant of Restricted Quintic"},"content":{"rendered":"
\n
\"Zero<\/a>

Zero Set of Discriminant of Restricted Quintic – Greg Egan<\/p><\/div>\n<\/div>\n

This image by Greg Egan<\/a> shows the set of points \\((a,b,c)\\) for which the quintic \\(x^5 + ax^4 + bx^2 + c \\) has repeated roots. The plane \\(c = 0\\) has been removed.<\/p>\n

The fascinating thing about this surface is that it appears to be diffeomorphic to two other surfaces, defined in completely different ways, which we discussed here:<\/p>\n

• Involutes of a cubical parabola<\/a>.<\/p>\n

• Discriminant of the icosahedral group<\/a>.<\/p>\n

The icosahedral group is also called \\(\\mathrm{H}_3\\). In his book The Theory of Singularities and its Applications<\/i>, V. I. Arnol’d writes:<\/p>\n

\nThe discriminant of the group \\(\\mathrm{H}_3\\) is shown in Fig. 18. Its singularities were studied by O. V. Lyashko (1982) with the help of a computer. This surface has two smooth cusped edges, one of order 3\/2 and the other of order 5\/2. Both are cubically tangent at the origin. Lyashko has also proved that this surface is diffeomorphic to the set of polynomials \\(x^5 + ax^4 + bx^2 + c\\) having a multiple root.\n<\/p><\/blockquote>\n

Figure 18 of Arnold’s book is a hand-drawn version of the surface below:<\/p>\n

\n
\"\\(\\mathrm{H}_3\\)<\/a>

\\(\\mathrm{H}_3\\) Discriminant – Greg Egan<\/p><\/div>\n<\/div>\n

Arnol’d’s claim that the discriminant of \\(\\mathrm{H}_3\\) is diffeomorphic to the set of polynomials \\(x^5 + ax^4 + bx^2 + c\\) having a repeated root is not literally true, since all such polynomials with \\(c = 0\\) have a repeated root, and we need to remove this plane to obtain a surface that looks like the discriminant of \\(\\mathrm{H}_3\\). After this correction his claim seems right, but it still deserves proof.<\/p>\n

Puzzle.<\/b> Can you prove the corrected version of Arnol’d’s claim?<\/p>\n

References <\/h2>\n

Arnol’d’s claim appear on page 29 here:<\/p>\n

• Vladimir I. Arnol’d, The Theory of Singularities and its Applications<\/i>, Cambridge U. Press, Cambridge, 1991.<\/p>\n

The following papers are also relevant:<\/p>\n

• Vladimir I. Arnol’d, Singularities of systems of rays, Uspekhi Mat. Nauk<\/a><\/i> 38:2<\/b> (1983), 77-147. English translation in Russian Math. Surveys<\/i> 38:2<\/b> (1983), 77–176. <\/p>\n

• O. Y. Lyashko, Classification of critical points of functions on a manifold with singular boundary, Funktsional. Anal. i Prilozhen.<\/a><\/i> 17:3<\/b> (1983), 28–36. English translation in Functional Analysis and its Applications<\/i> 17:3<\/b> (1983), 187–193<\/p>\n

• O. P. Shcherbak, Singularities of a family of evolvents in the neighbourhood of a point of inflection of a curve, and the group \\( \\mathrm{H}_3\\) generated by reflections, Funktsional. Anal. i Prilozhen.<\/a><\/em> 17:4<\/strong> (1983), 70–72. English translation in Functional Analysis and its Applications<\/em> 17:4<\/strong> (1983), 301–303.<\/p>\n

• O. P. Shcherbak, Wavefronts and reflection groups, Uspekhi Mat. Nauk<\/a><\/i> 43:3<\/b> (1988), 125–160. English translation in Russian Mathematical Surveys<\/i> 43:3<\/b> (1988), 1497–194.<\/p>\n

All these sources discuss the discoveries of Arnol’d and his colleagues relating singularities and Coxeter–Dynkin diagrams, starting with the more familiar \\(\\mathrm{ADE}\\) cases, then moving on to the non-simply-laced cases, and finally the non-crystallographic cases related to \\(\\mathrm{H}_2\\) (the symmetry group of the pentagon), \\(\\mathrm{H}_3\\) (the symmetry group of the icosahedron) and \\(\\mathrm{H}_4\\) (the symmetry group of the 600-cell).<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This image by Greg Egan<\/a> shows the set of points \\((a,b,c)\\) for which the quintic \\(x^5 + ax^4 + bx^2 + c \\) has repeated roots. The plane \\(c = 0 \\) has been removed. This surface is connected to involutes of a cubical parabola<\/a> and the discriminant of the icosahedral group<\/a>.<\/p>\n

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