About the theory of equations and the quintic.

Question

Maybe this is not the place for this question but I will try.

I was doing some research about equations and systems of equations because I am working right now with some identities that lead to some complicate systems and I was reading randomly in the web.

I found the wikipedia pages of the Bring radical used to solve the quintic and the page about "Theory of Equations". This last page says that the term "theory of equations" is nowadays mostly used in history of mathematics because algebra evolved beyond that after the development of Galois theory. Indeed, I was trying to find recent publications of researcg papers about these topics and I didn't find anything new and valuable, like if the field is abandoned. Maybe I am looking bad, but it seems that the interest in the theory of equations is low nowadays. My question is about what is the reason for that. Is it just that they are not fashion currently or it really happen that the field was completely closed and solved after Galois? I am a bit impressed that quintic and higher degrees equations are not still producing new ideas in any manner. After all, I don't think we trully understand that well the tools and intrincacies about these equations and there should remain something interesting and valuable to study there. Maybe it is just a certain change of name of the fields or so but I would like to know more about this phenomenum.

So, all in all, what are some open problems that properly classifies in the tag "theory of equations"?

Thanks in advance and sorry if this is not the place. And also I don't want to be unrespectful to people working in these fields, it is just that I cannot find you properly in the literature!

Answer 1

This is really a question for history of mathematics. In the nineteenth century, the fundamental theorem of algebra was rigorously proved as well as the criterion for finding roots of polynomials in terms of radicals. This was the culmination of the Theory of equations. As the Wikipedia article states:

Before Galois, there was no clear distinction between the “theory of equations” and “algebra”. Since then algebra has been dramatically enlarged to include many new subareas, and the theory of algebraic equations receives much less attention. Thus, the term "theory of equations" is mainly used in the context of the history of mathematics, to avoid confusion between old and new meanings of “algebra”.

One branch of the area is the efficient algorithmic numerical solution of linear and higher degree systems of equations. This is still actively being researched. A much different branch is the effective solution in integers of algebraic equations. An active and applied topic in this branch is elliptic curves. A much more extensive and abstract branch is algebraic geometry which has been actively researched in the last century through the present day.

Answer 2

Although the field might not be a very popular one, I would not say it is an abandoned one. I recommend the lecture of the wonderful piece of mathematics

Geometry of the quintic (2009), by J. Shurman.

Here is the conclusion of this book (Section 11: Onwards):

For the historical context of the first six chapters of this book, see Klein’s Development of mathematics in the 19th century [6, 7]. Meanwhile, mathematical research continues on these topics. For example, Dummit [5] shows how to solve a quintic by radicals when its Galois group is solvable, Buhler-Reichstein |2] generalizes Kronecker’s Theorem, Crass $[3, 4]^*$ discusses solving the sextic by iteration of more than one variable, and Beukers [1] addresses the diophantine equation $x^5 + y^3 = z^2$ using the icosahedral invariants.

Nonicosahedral approaches to the quintic abound, of course. See Sturmfels [8] for recent results on solutions by hypergeometric series, or more generally consult the extended bibliography accompanying the poster Solving the Quintic with Mathematica.

Finally, Klein’s original masterpiece [6], especially its second part, contains lovely material beyond this book that it inspired.

$(*)$ $\scriptsize{\text{I added the bib entry [4]}}$

[1] Beukers, Frits, The Diophantine equation $Ax^p+By^q=Cz^r$, Duke Math. J. 91 (1998), no. 1, 61--88.

[2] Buhler, J.; Reichstein, Z. On the essential dimension of a finite group. Compositio Math. 106 (1997), no. 2, 159--179.

[3] Crass, Scott Warren, Solving the sextic by iteration: A complex dynamical approach. Thesis (Ph.D.)–University of California, San Diego. (1996) 207 pp. ISBN: 978-0591-42221-4

[4] Crass, Scott. New light on solving the sextic by iteration: an algorithm using reliable dynamics. J. Mod. Dyn. 5 (2011), no. 2, 397--408.

[5] Dummit, D. S., Solving solvable quintics, Math. Comp. 57 (1991), no. 195, 387--401.

[6] Klein, Felix. Lectures on the icosahedron and the solution of equations of the fifth degree. Translated into English by George Gavin Morrice. Second and revised edition. Dover Publications, Inc., New York, N.Y., 1956. xvi+289 pp.

[7] Klein, Felix. Development of mathematics in the 19th century. With a preface and appendices by Robert Hermann. Translated from the German by M. Ackerman. Lie Groups: History, Frontiers and Applications, IX. Math Sci Press, Brookline, Mass., 1979. ix+630 pp. ISBN: 0-915692-28-7

[8] Sturmfels, Bernd. Solving algebraic equations in terms of $\scr A$-hypergeometric series. Formal power series and algebraic combinatorics (Minneapolis, MN, 1996). Discrete Math. 210 (2000), no. 1-3, 171-181.