Standing by for a correction to our view of the physical

I’ve spent a number of years using this blog to highlight the way that mathematical things seem to operate in very natural occurrences like the way our brains work, the way ants navigate, the way plants calculate an efficient consumption rate of their stored starch, the collective behavior of insect colonies, flocks, schools, and so much more. I do this to counter the view that mathematics is merely a miraculously productive human tool. I believe that if we look at it more carefully, we will see that mathematics itself is part of nature, as are all aspects of our thoughtful and imaginative lives. This is why I decided to bring a Quanta Magazine article to your attention today. Given the title ‘Mathematician Measures the Repulsive Force Within Polynomials,’ the article uses the language of physics to describe the behavior of numbers, a language that mathematicians use as well. Rather than finding mathematical action in a physical thing, this discussion finds physical behavior in a mathematical thing.

We are all familiar with repulsive forces in the physical world, like the repulsive force between two, like magnetic poles. In May 2020, Quanta Magazine reported on a result in mathematics, from mathematician Vesselin Dimitrov of the University of Toronto, where the proof is understood as a demonstration of what mathematicians call a ‘repulsion’ between numbers. Dimitrov proved a decades-old conjecture known as the Schinzel-Zassenhaus conjecture which concerns the roots of a particular family of polynomials. Kevin Hartnett begins by explaining mathematics’ version of physical repulsive actions:

When mathematicians look at the number line, they see the same type of trend. They look at the tick marks denoting the positive and negative counting numbers and sense a kind of numerical force holding them in that equal spacing. It’s as though, like mountain lions with their wide territories, integers can’t exist any closer together than 1 unit apart.

The spacing of the number line is the most basic example of a phenomenon found throughout the field of number theory. It crops up in the study of prime numbers and in the relationships between solutions to different types of equations. Mathematicians can better understand these important values by quantifying the force that acts between them. (emphasis added)

Let’s look at the result that is the subject of Hartnett’s article. If it’s been a long time since you’ve thought about polynomials, you may recall from Algebra classes that the roots of a polynomial equation y=f(x) are the x values that produce 0 for y. On the Cartesian plane, they are the x values where the graphed curve of the polynomial intersects the x axis. These intercepts mark the real roots (the real numbers that produce 0 when plugged into the polynomial), if they exist. Polynomials may also have roots that are complex numbers – numbers represented as the sum of a real number and some multiple of i which is the symbol for the square root of -1, known as the imaginary unit. Roots plotted on the complex plane will include roots that are complex numbers. Finding geometric relationships among the roots of a polynomial has long been a subject of study in mathematics. The Gauss-Lucas theorem, for example, establishes a geometric relationship, on the complex plane, between the roots of a polynomial and the roots of its derivative (which measures or quantifies the way y values change as x values vary for that polynomial). Derivatives of polynomials are also polynomials, and the theorem says that the roots of the derivative of the polynomial all lie within the smallest polygon, on the complex plane, that contains all the roots of the polynomial itself.

The result discussed in Quanta has to do with a particular class of polynomials called cyclotomic polynomials. These are polynomials, with integer coefficients, that are irreducible (meaning they cannot be factored), whose roots, on the complex plane, all lie on the unit circle (a circle centered at the origin with a radius of 1). There are an infinite number of such polynomials and there is a formula for producing them. It is striking that all of the roots of all such polynomials lie on this circle. The Quanta article discusses the proof of a conjecture about the relationship between the roots of these cyclotomic polynomials and non-cyclotomic polynomials.

In 1965, Andrzej Schinzel and Hans Zassenhaus predicted that the geometry of the roots of cyclotomic and non-cyclotomic polynomials differs in a very specific way. Take any non-cyclotomic polynomial whose first coefficient is 1 and graph its roots. Some may fall inside the unit circle, others right on it, and still others outside it. Schinzel and Zassenhaus predicted that every non-cyclotomic polynomial must have at least one root that’s outside the unit circle and at least some minimum distance away.

Or, to put the Schinzel-Zassenhaus conjecture in terms of repulsion, it predicted that the smallest roots of a non-cyclotomic polynomial — which might fall within the unit circle — effectively push other roots outside the unit circle, like magnets pushing each other away.

The minimum distance was expected to depend on the degree of the polynomial, specifically it was conjectured to be some constant number divided by the degree of the polynomial (or the power of the leading term). Dimitrov proved that this minimum distance is, in fact, (log2)/4d, where d is the degree of the polynomial. Log2 is a constant, and while the discussion allows for the possibility that the result could, perhaps, be tweaked to be something like (log3)/ 5d, the fact was established that the distance does depend on the quotient of a constant and a multiple of the degree of the polynomial.

One might say that these observations are just observations of the distance between numbers. But it’s more than that. These distances are produced by the numbers themselves, by their interaction in the polynomials. It is not unusual for mathematicians to talk about the ‘behavior’ of mathematical things – the behavior of solutions or, in this case, the behavior of roots. Is it a metaphor, or does this language emerge from an intuition about what a number really is? I suspect the latter is true. Numbers appear to be the names we have given to the elements of things we collect, or the duration of events. But within mathematics they have undergone a significant evolution, forcing us to examine other things, like the notion of a continuum, or the effects of an imaginary unit. Their geometric interpretation opened up whole new worlds of mathematical events. I bring the repulsion principle to your attention to make the point that the nature of mathematical things is just not very clear, and I am convinced mathematics doesn’t belong to us. Don’t misunderstand. I am meaning to be neither romantic nor mystical about these things. I mean to see something more clearly. A correction to our view of mathematics will bring with it a correction to our view of the physical world.

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