Semi-algebraic properties of Minkowski sums of a twisted cubic segment

Abstract

We find a semi-algebraic description of the Minkowski sum ${\mathcal{A}}_{3,n}$ of $n$ copies of the bounded twisted cubic $\{(t,t^2,t^3)\mid -1\le t\le 1\}$ for each integer $n\ge 3$. These descriptions provide efficient membership tests for the sets ${\mathcal{A}}_{3,n}$. These membership tests in turn can be used to resolve some instances of the underdetermined matrix moment problem, which was formulated by Michael Rubinstein and Peter Sarnak in order to study problems related to $L$-functions and their zeros.

19 pages, 3 figures, major revision