Back to basics, I want to talk about polynomials and roots.
When studying polynomials, mathematicians are especially interested in their roots, the values of a variable that make the polynomial equal zero. A polynomial has as many roots as its degree (the value of the largest exponent), so that $x^3 + 3 = 0$ has three roots, whilst $x^9 - 3x^2 + 4 = 0$ has nine roots.
An important amusing field of study is the one in which we try to find a relation between a root of a polynomials and each others. For example: when graphed, the roots of some polynomials fall exactly on the vertices of regular polygons (they stand apart by an exact geometric length). But there could be more sublet geometric relationships…
What kind of pattern can you get? Can you get ANY pattern?
Let’s focus, for the moment, only onto an important class of polynomials: the cyclotomic polynomials (CP). Those are the ones that cannot be factored into smaller ones, but you can use them to build polynomials. The first two ones, and the simplest ones, are $(x+1)$ and $(x-1)$. You cannot factor them into smaller polynomials, but you can use them (for example by multiplying them) to get ($x^2 - 1$).
The tenth cyclotomic polynomials is $x^4 - x^3 + x^2 - x + 1$ (want to have fun? Prove it!).
The roots of CP follow a very special geometric pattern. To see it, we need to start with the complex plane in which the $x$-axis plots real numbers, and the $y$-axis plots imaginary ones. Then inscribe a circle of radius $R = 1$ (no matter if cm, metres or whatever) around the origin and you get the famous Unit Circle: the roots of CP ALL lie on this circle.
They bear an elegant name: the Roots of Unity.
Seeking the Roots of Unity.
Polynomials consist of coefficients and variables raised to powers (see examples above).
Roots are the values of x that make the polynomial equal zero. The roots of a CP all lie on a circle with radius 1 unit, centred around the origin of the complex plane. These are the “Roots of Unity”.
You can admire the meaning of this just above.
The problem is that the most of polynomials are non cyclotomic, and their roots are not the roots of unity. This is the case with almost any combination of coefficients, variables and exponents you could come up with.
In $1965$, A. Schinzel and H. Zassenhaus predicted that the geometry of the roots of CP and non-CP differ in a very specific way: take ANY non CP whose first coefficient is $1$. Find his roots and graph them. Some may fall inside the unit circle, some right on it and others may fall outside it.
Schinzel and Zassenhaus predicted hence that EVERY non CP MUST have at least ONE root that falls outside the Unit Circle and at least some minimum distance away.
Another fascinating way to see this is in terms of repulsions: “the smallest roots of any non CP, which might fall within the Unit Circle, effectively push other roots outside the Unit Circle, like magnets pushing each other away
It’s like to think about the roots as negative electric charged particles that repel each other with a force that decays when the distance increases.
(Continues after the D.s. [Durante Scriptum])
D.s. I DARE thou to say that mathematics (and physics) is boring. "Shut up and calculate”. (R. Feynman).
Repulsive Roots.
So, according to Schinzel-Zassenhaus conjecture, every non CP must have at least one root that is at least some minimum distance outside the Unit Circle. That distance varies depending on the values of the largest power in the polynomial.
In the above example, the black dots are the roots of
$$x^{7} + 2x^{5} - 12x^{4} - 12x^{3} + 2x^{2} + 1 = 0$$
The conjecture’s main prediction has the feel of a physics equation. It says every non CP should have at least one root that is outside the Unit Circle by a distance equal to a constant number divided by the degree of the polynomial.
If we had a non CP of degree $23$, the conjecture predicts that is should have a root at least $\dfrac{1}{23}$ of unit outside the circle.
It’s a powerful statement, but for decades only weaker forms of this conjecture has been managed to be proven. Indeed, Schinzel and Zassenhausem themselves managed only to prove that every non CP has a root at least $\left(\dfrac{1}{4}\right)^d$ [where $d$ = degree of the polynomial] outside the Unit Circle, a much smaller distance than the conjectured one.
Many improvements have been made, yet the conjecture still remains unsolved. Often, when a prominent math problem remains open for a long time, it’s because mathematicians simply lack the technique to solve it. Dream as you might of flying to the moon, you’re not getting there until someone invents a rocket.
And then it comes V. Dimitrov: he transformed a question about the size of roots of polynomials into a question about the size of the values associated to a related but different type of mathematical object called power series (it’s like a polynomial, only with infinitely many terms).
Want to know more and really go deep? Have fun at ArXiv.org “A proof of the Schinzel-Zassenhausen conjecture on polynomials”. It’s really elegant and well understandable!
Click here for the article
