The Accidental Astrophysicists
Polynomial equations with conjugates: how many roots?
Albert Einstein is, well, Albert Einstein. He is perhaps the best known, most recognizable, iconic scientist of all time. He once said,
Do not worry about your problems with mathematics, I assure you mine are far greater.
Today I want to talk about some mathematical problems we have. In particular the problem of solving equations—equations of a type that are not as well known as polynomial equations.
There is something fundamental about solving equations. Einstein faced the problem of solving equations, especially in his theory of General Relativity. Many early results in mathematics were driven by the desire to solve an equation of one kind or another. Even simple linear equations were hard for early mathematicians: consider the equation,
For us the solution is clear, , but negative numbers were not accepted as “real” numbers initially. See the fun book by Alberto Martínez on Negative Math for a history of negative numbers.
The Greeks had even more difficult times with quadratic equations: what is the solution to
The root is irrational, again a “real number” for us, but not apparently for the early Greek mathematicians.
For equations of one variable, in modern terms, there is the famous Fundamental Theorem of Algebra (FTA):
Theorem: Any polynomial equation with
where the coefficients are complex numbers, has a solution over the complex numbers.
This ends the story of equations in one variable—or does it? Not quite. There are countless kinds of more-general equations to consider, indeed many generalizations of the notion of “polynomial” itself. I will discuss several such generalizations.
The FTA has been generalized to include more general functions of all kinds. For what classes of functions is the equation
guaranteed to have a solution over the real or complex numbers? Note that we need to have some restriction on in order to get a reasonable theory.
One interesting family of functions are the harmonic functions. We can form some of them as polynomials in variables and their complex conjugates . For the case of two variables with complex coefficients, this in fact gives all the harmonic polynomials. Thus for an ordinary (analytic) complex polynomial we can form the harmonic polynomial . Thus, questions arise like: how many roots can the equation,
have? The theory here is still incomplete, because the behavior of such equations is much more involved than the behavior of pure polynomials. Yet there are some quite pretty results that are known, and there are many interesting open questions.
Dmitry Khavinson and Genevra Neumann have written a general introduction to this wonderful theory here. Right away the behavior of such polynomials is much more complex. For example, each of the harmonic polynomials
has an infinite number of solutions. The equation holds if and only if is a real number, so the solution set is the star formed by equally spaced rays from the origin.
Given such infinite cases, it is remarkable that one can find broad conditions under which there are finitely many distinct zeroes—though still more than the degree as with ordinary polynomials. There is the following pretty theorem due to Alan Wilmshurst:
Theorem: If is a harmonic polynomial of degree such that , then has at most zeroes.
The intuition behind this is to look at the behavior of the real and the imaginary parts of separately. They have zero sets that are lines. Of course both the real and the imaginary parts must both be zero to have a zero of , so the issue is how many intersections can there be? This is where the comes from. The black lines are where the real part is zero, and the red lines where the imaginary part is zero.
Polynomials Plus Plus
Khavinson and Neumann proved a related result concerning rational harmonic functions—polynomials plus plus. They proved:
Theorem: Let , where and are relatively prime polynomials in , and let be the degree of . If , then the number of solutions of
is at most .
It turns out that the mathematical upper bound is closely related to the physics concept of gravity lenses. They have been called accidental astrophysicists. I will explain why in the next section.
According to Einstein’s theory of General Relativity light is bent in the presence of gravity. This leads to the phenomena of gravity lenses. A single object in the presence of a powerful source of gravity can appear as multiple objects. Here is a picture of the formation of such images:
One of the critical questions is how many images can be formed? This is of importance to astrophysicists.
The surprise is it possible to model the number of images not with a polynomial equation, but with a harmonic equation. Actually, the astrophysicist Sun Rhie proved a lower bound, and she conjectured that her construction was optimal. Thus, when Khavinson and Neumann proved an upper bound, they were actually solving her conjecture. Apparently they did not know this connection before they proved their theorem. Science, especially mathematics, can be surprising—there are often unforeseen connections. Thus the name “accidental astrophysicists.”
There are many complexity questions that arise from these theorems. Can we decide efficiently how many roots a harmonic polynomial has? Also can we find them—at least approximately? Also are there applications to complexity theory of these results?