Congratulations to Mathematics student Lola Thompson, who was one of four winners of the Graduate Poster Session held recently at the Top of the Hop! Enjoy your winnings, Lola! (Read on for a summary of Thompson’s poster.)
We are interested in studying polynomials that have a particularly simple form: f(x) = x^n – 1; where n ranges over all of the positive integers. In particular, we would like to answer the following natural question: “How often does x^n – 1 have a divisor of every degree between 1 and n?” We can prove that polynomials with this property are very rare. In fact, if you throw all of the positive numbers into a paper bag and draw one out at random, the probability that the corresponding polynomial has a divisor of every degree is 0.
However, there are still infinitely many integers n for which the corresponding polynomial does have a divisor of every degree between 1 and n, so these n’s live in a sparse-but-infinite set. Although it is interesting to note that these polynomials are rare, we would like to be able to say even more. We would like to quantify exactly how rare it is for x^n – 1 to have a divisor of every degree between 1 and n by modeling its frequency with an explicit function. As it turns out, we can show that the function X/logX is a good model for the frequency of integers whose corresponding polynomials have this property. Our proof of this theorem uses a technique called sieving.
Just like we can sift out sand at the beach so that we are left with only shells and rocks, we can sift out the numbers n for which x^n – 1 does not have a divisor of every degree; this leaves us with only the numbers that do have this property. This type of work may have some applications in the “real” world; information about the factorization of polynomials is often used in cryptography. In addition, we are hoping to apply some of the techniques used in our work on polynomials in order to answer statistical questions about elliptic curves, which are also useful in digital security.
summary by Lola Thompson