Engineering Systems and Free Semi-Algebraic Geometry

  • Mauricio C. De OliveiraEmail author
  • J. William Helton
  • Scott A. Mccullough
  • Mihai Putinar
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 149)


This article sketches a few of the developments in the recently emerging area of real algebraic geometry (in short RAG) in a free* algebra, in particular on “noncommutative inequalities”. Also we sketch the engineering problems which both motivated them and are expected to provide directions for future developments. The free* algebra is forced on us when we want to manipulate expressions where the unknowns enter naturally as matrices. Conditions requiring positive definite matrices force one to noncommutative inequalities. The theory developed to treat such situations has two main parts, one parallels classical semialgebraic geometry with sums of squares representations (Positivstellensatze) and the other has a new flavor focusing on how noncommutative convexity (similarly, a variety with positive curvature) is very constrained, so few actually exist.


Linear Matrix Inequality Positive Semidefinite Free Algebra Symmetric Polynomial Weyl Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Mauricio C. De Oliveira
    • 1
    Email author
  • J. William Helton
    • 2
  • Scott A. Mccullough
    • 3
  • Mihai Putinar
    • 4
  1. 1.Mechanics and Aerospace Engineering DepartmentUC at San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  3. 3.Department of MathematicsUniversity of FloridaGainesvilleUSA
  4. 4.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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