Many models in science and engineering are described by polynomials.  Computational algebraic geometry gives tools to analyze and exploit algebraic structure.  In this talk, we offer a user-friendly introduction to some of these notions, including dimension (formalizing degrees of freedom), degree (formalizing the number of solutions to a polynomial system) and 0-1 laws in algebraic geometry (solution sets to polynomial systems exhibit similar behavior for all but a measure 0 subset of problem instances).  We will also mention algorithms, based on Gröbner bases (symbolic techniques) and homotopy continuation (numerical techniques).Applied examples are drawn from the structure-from-motion problem in computer vision, where the task of building a 3D model from multiple 2D images leads to nontrivial polynomial systems.
References include:
J. Kileel, Minimal Problems for the Calibrated Trifocal Variety, SIAM Journal on Applied Algebra and Geometry 1 (2017) 575-598.
J. Kileel, Z. Kukelova, T. Pajdla and B. Sturmfels, Distortion Varieties, Foundations of Computational Mathematics, first online.