Resources

Textbooks

Nice introductory textbooks guiding many aspects of nonlinear algebra that I found/read/studied. I still read these occasionally and when I conduct a reading session with students I try to use these books as an initial step. In many univerisities, the libraries provide free online access to some of these textbooks.
• Ideals, Varieties, And Algorithms by David A. Cox, John Little and Donal O'Shea
  

• Using Algebraic Geometry by David A. Cox, John Little and Donal O'Shea
  

• Introduction to Tropical Geometry by Diane Maclagan and Bernd Sturmfels
  

• Invitation to Nonlinear Algebra by Mateusz Michałek and and Bernd Sturmfels
  if you are a person who prefers shorter reading, there is a nice expository article about nonlinear algebra
• Matroid Theory by James Oxley
  anything shorter than this? there is also a nice article about a matroid
• The Numerical Solution of Systems of Polynomials Arising in Engineering and Science by Andrew J. Sommese and Charles W. Wampler
  

• Lectures on Polytopes by Günter M. Ziegler
  

• A Course in Convexity by Alexander Barvinok
  
  
  

  Introductory level articles

  These are research papers all examine various aspects of nonlinear algebra. Most of them are papers that I
  1) enjoyed reading during my earlier years of my graduate studies,
  2) found useful to study an introduction of a specific topic or,
  3) think worth to read because they are cited frequently.
  Many of these papers are available at arXiv.
  Caveat : The meaning of "introductory level" can be (seriously) personal and interpreted in various way ;)

• A Polyhedral Method for Solving Sparse Polynomial Systems by Birkett Huber and Bernd Sturmfels
  

• Algebraic Matroids in Action by Zvi Rosen, Jessica Sidman and Louis Theran
  

• Computing the multiplicity structure in solving polynomial systems by Barry H. Dayton and Zhonggang Zeng
  

• Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems by Jonathan D. Hauenstein and Frank Sottile
  

• The Bergman complex of a matroid and phylogenetic trees by Federico Ardila and Caroline J. Klivan
  

• Algebraic matroids and set-theoretic realizability of tropical varieties by Josephine Yu
  

• Algebraic Degree of Polynomial Optimization by Jiawang Nie and Kristian Ranestad
  

• What is a matroid? by James Oxley




Software

• Macaulay2
  

• HomotopyContinuation.jl
  

• Bertini
  

• alphaCertified
  
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