Real Spectrum and the Abstract Positivstellensatz

Algebraic geometry emerged from the study of subsets of C defined by polynomial equations, namely algebraic sets. Parallel to this, real algebraic geometry emerged from studying subsets of R defined by polynomial equations and inequalities, namely semi-algebraic sets. Like algebraic geometry, classical methods in real algebraic geometry provided useful but limited intuition on the relationship between algebraic properties of polynomials defining a semi-algebraic set and geometric phenomena of the set. This sparked the development of a real algebraic analog of the prime spectrum, the real spectrum, which installed a deeper understanding of the influence real algebra and real geometry have on one another. This paper studies the real spectrum of commutative rings, focusing on how it is used to develop a tighter bond between algebraic properties of rings, and geometric properties of semi-algebraic sets. We begin by recalling some classical real algebraic theory, leading up to the key theorem relating semi-algebraic subsets of R and algebraic structures in R[x1, . . . , xn]: the Positivstellensatz. We use the Positivstellensatz as a motivator for constructing the real spectrum, in order to have a more complete algebraic understanding of semi-algebraic sets defined by non-strict inequalities. Finally, we discuss a generalization of the Positivstellensatz, the Abstract Positivstellensatz, to general real spectra (beyond R). Besides being of intrinsic interest, the sophisticated algebraic study of semialgebraic sets has had many successful applications throughout mathematics. One particular field is optimization. In particular, polynomial optimization is concerned with maximizing or minimizing polynomial functions over semi-algebraic sets. Understanding the feasible region of these optimization problems is precisely understanding the geometry of semi-algebraic sets. Many authors have used real algebraic geometry to advance the theoretical and complexity theoretic understanding of such optimization problems (see [3]).