Real algebraic geometry for matrices over commutative rings

Abstract

We define and study preorderings and orderings on rings of the form $M_n(R)$ where $R$ is a commutative unital ring. We extend the Artin-Lang theorem and Krivine-Stengle Stellensätze (both abstract and geometric) from $R$ to $M_n(R)$. While the orderings of $M_n(R)$ are in one-to-one correspondence with the orderings of $R$, this is not true for preorderings. Therefore, our theory is not Morita equivalent to the classical real algebraic geometry.

References in corpus (1)

• Formally real involutions on central simple algebras

Cited by in corpus (14)

• Moment problems for operator polynomials
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• Positive trace polynomials and the universal Procesi-Schacher conjecture
• There are many more positive maps than completely positive maps
• Positive univariate trace polynomials
• A Real Nullstellensatz for Free Modules
• Finsler's Lemma for Matrix Polynomials
• Some Positivstellensätze for polynomial matrices
• A Matrix Positivstellensatz with lifting polynomials
• Finite convergence of the Moment-SOS hierarchy for polynomial matrix optimization
• Matrix Fejér-Riesz theorem with gaps
• Handelman's Positivstellensatz for polynomial matrices positive definite on polyhedra
• Sum-of-square-of-rational-function based representations of positive semidefinite polynomial matrices
• Some applications of Scherer-Hol's theorem for polynomial matrices