Polynomials, Dynamics, and Choice

There is a wealth of depth in even the simplest-seeming mathematical questions.  One nice example of this is the familiar Newton-Raphson method of root approximation often taught in first-semester calculus.  On its face, it's a fairly straightforward application of the derivative and one which is highly effective at quick convergence.  Pick a guess, apply, and get your zero.  Nice and simple.

 

But as one expects, I wouldn't have mentioned this example if there weren't a lot of subtleties.  Quite aside from some initial guesses leading to a zero derivative and breaking the mechanism, the question of which initial value leads to which root (and how long it takes) is a well-known first example of chaotic behavior.  The boundaries between regions exhibit beautiful fractal patterns.  While you can no longer submit your own examples to David Joyce's "Newton Basin" generator, his website still has many gorgeous images (and is still the first Google hit for this topic); Github shows plenty of people who have written their own versions of such software as well.

 

The book under review also has lush graphics of this type (see Sections 5.4 and 5.5), which is appropriate for a book on root-finding in polynomials.  Similarly, in its treatment of the algebraic approach to finding roots, there is very helpful color-coding of symmetries of symmetric polynomials (p. 41) and the cubic formula (p. 49), and in general, has many clear diagrams of things like permutations and dodecahedral symmetry (relevant for Abel's nonexistence result for a quintic formula).

 

Unfortunately, what's not clear at all is the book's intended audience.  It's certainly not a textbook, nor does it claim to be one. Though it covers many attractive ideas of symmetry, carefully develops the quadratic and cubic formulas, and introduces one to the complex dynamics of both Newton's and Halley's method (the latter of which I was glad to learn about!), not only are there no exercises, but the exposition is extremely conversational to a point where it would be difficult to pin down exactly where a given result was finished.  The terminology is also intentionally non-standard - "combinatorics" and "permutation" are relegated to footnotes, while "shuffle" is the standard term here.

 

But, in my estimation, the book is still far too "mathematical" for what seems to be its intended audience of the legendary educated member of the laity.  Very lucid sections that caringly take a reader through big ideas are followed by ones that unsuccessfully attempt to make difficult ideas intuitive.  Figures 5.9-5.11, and the surrounding material, are an example of this.  Even after several readings, I'm still not sure exactly how to define a critical point or immediate basin of attraction (in a dynamical system) properly, nor what picture I should have in my mind for the former.  I also couldn't even find "critical point" in the index until I looked at the fourth item under "dynamics", and even then had to search the referenced page diligently, since there are very few numbered or labeled items, presumably intentionally.

 

The reader will notice that in the three words of the main title, "choice" has yet to make an appearance in this review.  In fact, the book is thoroughly framed and suffused by this word, because there is a powerful metaphor involved which the author repeatedly returns to.  To paraphrase him, we can try to first compute all our options, and only then make our choice to break the symmetry (such as the infamous "plus or minus" in the quadratic formula).  Or, we can simple make the choice (many choices simultaneously?) first and see where it leads - whether by computing Newton basins, or in using simulated agent-based modeling for other problems (see Sections 8.2 and 8.3).  It's a neat idea to use math for an analogy about all human choices, one which could stimulate lots of late-night dorm conversations (or final-exam-grading-avoidance daytime conversations) over its relevance or usefulness.

 

It's really a shame because both the extended metaphor and the new approach to the polynomial material deserve a hearing.  As further examples, I feel like I understand the Tschirnhaus approach to simplifying polynomials (and why Abel's proof would work) better now, and feel like I'd like to try an evolutionary approach to building Turing machines as well.  But trying to do everything at once, in the same book, without the usual mathematical structure, garbles the mathematical points while still leaving the material too difficult for a novice to digest.  I look forward to separate articles or short monographs on each, and can think of several journals (MAA and otherwise) which would be interested in that.

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Karl-Dieter Crisman is Professor of Mathematics at Gordon College.