Real algebraic geometry

Summary

Real algebraic geometry studies the sets of real solutions of systems of polynomial equations and inequalities, and the polynomials that are nonnegative on them. Its basic objects are real algebraic sets and semialgebraic sets over the real numbers or over a real closed field. Its main tools are the Tarski–Seidenberg theorem, orderings of fields, and certificates of positivity such as the Positivstellensatz. It has applications to optimization, robotics and the topology of real curves and surfaces.