The Wrath of Con(ceicao)

Research in the field of mathematics does not have fancy labs or fancy procedures like other fields. Unlike how the name of this post suggests, math research is neither stressful nor aggravating. While mathematics research does sometimes involve staring at equations erratically scrawled on a chalkboard, our tools are not limited to mundane methods like chalkboards and paper. Research in mathematics is a puzzle. It is finding trends and proving them. It is making relations between an unknown solution and known theorems to further knowledge of both the known and unknown.

This summer, Dr. Conceicao is working with two math majors: Sam VanFossen and Rachael Kelly, whom you can see busy at work in the pictures at the bottom of the page. Our research is based off of the Markov equation x^2+y^2+z^2 = xyz. The solution (x, y, z) to the equation is known as a triple. All integer solutions to Markov’s equation have been known for over 100 years using a simple recursive formula that allows a single solution to “branch” to other solutions. We, however, took a different approach to this equation. What if x, y, and z were polynomials, not integers? More specifically, what if they were monic polynomials? A monic polynomial is a polynomial whose leading coefficient is 1. For example, 5t^2 + 3t + 2 and t^2 + 3t + 2 are both polynomials but only the latter is monic.

Now, in order for polynomials to satisfy Markov’s equation, we must work in what is called a finite field. The “standard” field has infinite numbers. 1, 2, 3, 4… and so on. We can count upwards indefinitely. In finite fields, however, there are a set, or finite, number of elements. In our research, we work with primes of 1 mod 4. The “mod” of a number is the remainder of that number. Thus, 5 mod 4 is equivalent to 1 mod 4. We require our primes to be 1 mod 4 because -1 is a quadratic residue of numbers 1 mod 4. That is to say that i exists in the finite fields of primes that are 1 mod 4. i is the number such that i^2 = -1. In the standard field, i is known as an imaginary number because it does not exist; however, it is possible for i to exist in finite fields. In mod 5, -1 is equivalent to 4, since both -1 and 4 are divisible by 5 when 1 is added to them, and, as we know, 4 = 2^2. Thus, in mod 5, 2 = i. Further, 3^2 = 9 = 4 in mod 5. Therefore, both 3 and 2 are i in mod 5. exists in all finite fields created by numbers that are equivalent to 1 mod 4 . For example, 5, 13, 29, and 37 are primes that are equivalent to 1 mod 4.

Using this information and returning to Markov’s equation, we can prove that all monic solutions stem from a triple in the form (f, f + \beta, f^2 + \beta f - 2) where \beta = 2i and f is some monic polynomial. Thus, we have an infinite number of primes with an infinite number of trees with an infinite number of entries on each tree. And we aren’t even considering nonmonic solutions. Now that we have effectively found all solutions, what can we say about them? What other relationships do they have? A generating function is able to generate polynomials down a set “branch” of the tree. Using this function, we are able to have an explicit formula for the polynomials.

A classic conjecture that we would like to investigate is about the uniqueness of Markov polynomials. Does every polynomial appear on each tree as the maximum in a triple exactly once? Another question that we would like to investigate is if all Markov polynomials are reducible (or factorable). We are trying to understand the subfamily of Markov polynomials, F_n, defined by the generating function. Further, we can use the generating function to determine the values of n for which F_n has roots modulo a prime. For example, i will always be a root of F_3 and F_n if n=ko+3, for some integer o and all integers k. The value of o is known as its order. Rewriting the generating function as a Pell equation is useful in finding the order of certain roots. As such, we can create parallels between the Pell numbers, divisibility and Markov polynomials.  For example, in the finite field with p elements the order of β, or 2i, is the index of the first Pell number divisible by p. Similarly, the order of i for a specific prime p is found using Fibonacci numbers.

To further our research and save hundreds of hours, we use a program called Sage (when/if it works) to perform lengthy (to say the least) computations for us. We then enter the data on Excel spreadsheets, where we look for patterns in the data. When we find patterns, we then look for ways to prove that these patterns occur in every possible case or ways to generalize these results. Researching this topic has been a rewarding and illuminating experience, and we look forward to discovering more about these fascinating polynomials!

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