Unique minimal forcing sets and forced representation of integers by quadratic forms

Authors

Tahseen Rabbani, University of Virginia

Abstract

For subsets of natural numbers S and W, we say S forces W if any integer-matrix positive definite form which represents every element of S over the integers also represents every element of W over the integers. In the context of a superset S*, S is referred to as a unique minimal forcing set of W if for any subset S0 of S*, we have that S0 forces W if and only if S is a subset of S0. In 2000, Manjul Bhargava used his own novel method of “escalators" to prove the unique minimal forcing set of the natural numbers is T={1, 2, 3, 5, 6, 7, 10, 14, 15}, which was a refinement of the celebrated Conway-Schneeberger Fifteen Theorem. We use Bhargava's theory of escalators to develop an algorithm which determines whether a positive integer, interpreted as a singleton in the natural numbers, has a unique minimal forcing set within T and to establish several infinite families of positive integers without unique minimal forcing sets in T.

Author Bio

Tahseen Rabbani graduated from the University of Virginia in 2015, where he majored in mathematics. He is particularly interested in interactions between abstract algebra and linear algebra. This project was conducted under the supervision of Dr. Andrew Obus. Tahseen is currently working as a software developer in Madison, WI and plans to eventually attend a graduate program in mathematics. He is an avid movie buff who enjoys playing soccer and trying out new restaurants.

Faculty Sponsor

Andrew Obus

Recommended Citation

Rabbani, Tahseen (2016) "Unique minimal forcing sets and forced representation of integers by quadratic forms," Rose-Hulman Undergraduate Mathematics Journal: Vol. 17: Iss. 1, Article 1.
Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss1/1