The Mahler measure and its areal analog for totally positive algebraic integers

Using the method of explicit auxiliary functions, we first improve the known lower bounds of the absolute Mahler measure of totally positive algebraic integers. In 2008, I. Pritsker defined a natural areal analog of the Mahler measure that we call the Pritsker measure. We study the spectrum of the absolute Pritsker measure for totally positive algebraic integers and find the four smallest points. Finally, we give inequalities involving the Mahler measure and the Pritsker measure of totally positive algebraic integers. The polynomials involved in the auxiliary functions are found by our recursive algorithm.