Frobenius Problems on Polynomials

I heard this was the first time that X-SIG summer research had students from Mathematics Department. I am super happy that we finally have some mathematics going on in the summer.

In this project, we focus on Frobenius Problem on Polynomials. Like most mathematics research, our work is mostly to prove things, lemmas, propositions, and theorems. This topic is entirely original though there are tons of already established results in the classical Frobenius Problem in the number theory. Our work starts with some results in the classical Frobenius Problem. In the number theorem, we care about integers. In our problem, we replace integers with polynomials. Given some positive integers, the classical Frobenius Problem asks for the greatest integer that cannot be written as a nonnegative combination. In our problem, we ask for the smallest integer for which any polynomial whose degree is greater than that integer is representable by the given polynomials. In general, we want to understand better how polynomials behave.

Li and I usually start a new day with summarizing the previous work on our own. Then later at 10:30, we meet with our sponsor, Prof. Conceicao. We would present the results(proofs) of the previous day if any. If we end up having no progress, we would think through the problem together, rethinking about the problem, trying new strategies of proving or falsifying conjectures. Sometimes we let our computer program check some cases of the conjecture so that we don’t end up striving to prove something that is not true. Li and I type up our results in LaTex at the end of the day, which is also time-consuming. So a day mostly consists of thinking and discussing.

Staying at Gettysburg College in the summer is peaceful. The campus is more tranquil than it is in regular semesters because people stay on campus for a reason when there are a lot of ways to spend the summer. Living with people who understand each other’s commitment is a pleasant experience. At the same time, the dining service on campus is closed, and I have the time to cook for myself, so I practice my cooking skills a lot. I also enjoy cooking and baking for other summer residents. I will update pictures of our office and after-work activities when I have any.

Yiran

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