As winter approaches and fall splatters the treetops with 600nm-wavelength light, it is wise, albeit cold, to get out with a rake or a spade to catch the best of the last of warm sun rays. If, however, we are determined to shirk the changeable weather and stay inside with a computer in tow, the complex plane like the outdoor garden will offer plenty by way of uncovered roots, measured out squares, and eddies of discovery.
Let’s set for ourselves, before we begin, some fences we will attempt not to scale for the time being. To start, let us agree that the coefficients, at least, of our chosen polynomials shall be in the reals: a polynomial in $\mathbb{C}[z]$ will be mentioned but not studied in detail. Furthermore, let’s agree to focus our attention on the cubics: they will illustrate our point well enough, and will be simple enough to generalise: we have heard too much in various classes of the doings and dealings of the quadratics, but the cubics — the cubics retain some air of mystery, and will offer, I promise, a sufficient degree of fascination to keep us engaged with the task ahead. So, onwards! Armed with a graphing software, enthusiasm, and knowledge, let’s dig in!
Dramatis polynomiae
$x^{3}-5x^{2}+17x-13$: a real family man.
$x^{3}-5x^{2}+17x-\lambda$: his entire family.
$a_n z^n + a_{n-1} z^{n-1} + … + a_2 z^2 + a_1 z + a_0 $: ever a lady unwilling to commit to the numbers, scary in the full brunt of her generality.
Dress as the cubic formula for Halloween & scare all your learned friends!
A cubic would, via the Fundamental theorem of algebra, have three roots, and always three roots. One could look in ancient textbooks to find the thorny cubic analogue of a quadratic formula, but with modern calculators in hand, the monster of a formula is hardly useful for anything but making your stomach churn. Say we have found the roots by turning to a higher computational power. With these three roots in hand, and desperate for more, we can acquire more by taking the derivative of our polynomial, and finding its roots in turn. Here, even with cubics, we hit upon a marvellous pattern.
The first observation to make is: the roots of the polynomial’s derivative always seem to lie in the convex hull of the polynomial’s roots. This is the Gauss-Lucas Theorem, first implied in a quick note of Gauss’ in 1836 and proved some years later by Édouard Lucas.
We can get more by noticing that the centroid of our polynomial’s roots is the centroid of its derivative’s roots, and the centroid of its double derivative’s roots, and so on and so forth until you zero in on a point. For cubics, the roots form a triangle with a vertex on the real axis, and there is much more mathematics hiding in each triangle, waiting to be uncovered.
We jump first to some geometry. The inellipse of a triangle lies within and touches the triangle’s sides at the midpoints. (The prefix “in-” here is more akin to that of an incircle than to that of inexcusable.) As a mater of fact, it’s the maximal (in area) ellipse that can be inscribed in a given triangle. In 1881, Steiner showed that the foci of the inellipse inside the triangle of a cubic’s roots are exactly the roots of the cubic’s derivative! This inellipse now bears his name. So if we take a family of cubics with the same derivative, the foci of their Steiner inellipses will coincide.
Roots of the $x^{3}-5x^{2}+17x+\lambda$ family.
Let’s invite our cubic for a demonstration. We consider the whole family, $x^{3}-5x^{2}+17x+\lambda$, $\lambda\in\mathbb{R}$, of the relatives and relations of our friend $x^{3}-5x^{2}+17x-13$. The derivative of each such polynomial is the same. So the foci of the Steiner ellipse must be the same, and moreover, the centroid and the foci of the Steiner inellipses are invariant under $\lambda$. What’s more, we can construct a line of best fit of the three roots in the plane, and that’ll remain unchanged!
I’ve included some of the family in the drawings to the right and below, with and without their triangles. You can see them dancing around a single centroid and line of fit, with polynomial $x^3 -5x^2 +17x -19$ coming close, with roots roughly: $1.65, 1.67\pm 2.94$.
The roots moving in their hyperbolic tracks, the foci of the Steiner ellipse stand firm with the changing constant.
More roots, more glory
As you can imagine, the higher the degree of polynomial we start with, the more impressive our laser-focus on the spot that will be its last derivative’s final roots becomes. So, perhaps, if we are to impress our mates over a pint of beer a greater degree, both of the alcohol and the polynomial may well be required. And, after all, the thing looks pretty. Above is our friend $x^3 -5x^2 +17x -13$ along with some of his integrals, all having the same-root centroid in the complex plane.
Probably time to stop
For the green thumbs in the audience, there are always greater flowerbed arrangements to be built and more elegant ideas to be planted. I advise those interested to lasso in some probability theory and answer the query:
suppose we have an $n^{th}$-degree polynomial, $a_n z^n + a_{n-1} z^{n-1} + … + a_2 z^2 + a_1 z + a_0 $, where $n \rightarrow \infty$, and say that the $a_i$ are each randomly chosen: where do we expect to see the roots of this polynomial? We can grow for ourselves a formula for the expectation, a probability cloud valid for $n=3$, and it is interesting what it would condense into as $n$ grows… but this is really taking a leaf blower to the yard and ditching the spade, so we will talk no more of it here. For the answer, see reference 3.
As you can see, there are many avenues left unexplored. Thus you can spend all your time generating spades of polynomials, testing their properties and cackling in your infinite marvel, but perhaps let’s take a break and breathe some of that autumn air before it turns into the crisp of winter.
References
- D. Minda, S. Phelps, Triangles, ellipses, and cubic polynomials, The American mathematical monthly, 115, October 2008.
- M. Roth, Permutations given by polynomials, a talk given at Queen’s math club (January 26, 2017).
- L. Shepp, R. Vanderbei, The complex zeros of random polynomials, Transactions of the American mathematical society, vol. 347, No. 11, November 1995.
- J. Walsh, On the location of the roots of the derivative of a polynomial, Annals of mathematics, vol. 22, no. 2, December 1920.
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