A characterization of the algebraic degree in semidefinite programming

TL;DR

This paper characterizes the algebraic degree in semidefinite programming using symmetric polynomial theory, simplifying previous results and providing a new algebraic perspective.

Contribution

It introduces a novel characterization of the algebraic degree in SDP via coefficients of doubly symmetric polynomials, connecting algebraic geometry with symmetric polynomial theory.

Findings

Expressed algebraic degree as a monomial coefficient in symmetric polynomials

Simplified derivation of known results by Nie, Ranestad, and Sturmfels

Provided new insights into the algebraic structure of SDP solutions

Abstract

In this article, we show that the algebraic degree in semidefinite programming can be expressed in terms of the coefficient of a certain monomial in a doubly symmetric polynomial. This characterization of the algebraic degree allows us to use the theory of symmetric polynomials to obtain many interesting results of Nie, Ranestad and Sturmfels in a simpler way.