Mathematics & Statistics Colloquium: Degree D Points on Curves

Join the Department of Mathematics & Statistics for the 2025 Colloquium talk "Degree D Points on Curves" by Lea Beneish, assistant professor in the Department of Mathematics at the University of North Texas. This talk considers the question “What is the set of degrees D such that equations of the form f(x,y,z) = 0 have solutions of degree D?” This set turns out to have some interesting properties. 

The solution set of a polynomial in three variables in which every monomial has the same degree is referred to as a plane curve. A famous example of such a plane curve is given by the equation relating Pythagorean triples, $x^2+y^2=z^2$ where positive integer solutions correspond to right angled triangles with integer side lengths; in this case, there are infinitely many (primitive) solutions. However, an equation where two squares sum to the negative of another square, for example, $x^2+y^2=-z^2$ has no rational solutions, as the sum of two squares cannot be negative. Even though this equation has no rational solutions, it may have infinitely many solutions involving the imaginary number i. Since i has a minimal polynomial of degree 2 one says that such a point has degree 2 (for example, (i, 0, 1)) is a point of degree 2, whereas a rational point is considered a point of degree 1). This talk is based on joint work with Andrew Granville.

For more information, contact the colloquium chair, Dr. Michael Kerckhove.