Algorithms in real algebraic geometry

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Finding all Nash equilibria of a finite game using polynomial algebra (2006, methodology): “To solve a polynomial system by the homotopy continuation method, we create a family F of polynomial systems f1t D 0; f2t D 0; : : : ; fmt D 0 parametrized by a variable t lying in [0; 1 ], such that the polynomial system we want to solve is f11D 0; f21D 0; : : : ; fm1D 0, and the polynomial system”
Finding All Real Points of a Complex Curve (2006, background): “Symbolic algorithms along the same general lines exist, e.g., [ 1 , 3, 9].”
A semi-algebraic version of Zarankiewicz's problem (2014, background): “The Milnor-Thom theorem (see [5, 33, 42]) bounds the number of cells in the arrangement of the zero-sets V1, .”
Erdős-Hajnal-type Results on Intersection Patterns of Geometric Objects (2008, background): “We say that the description complexity of such a set S is at most κ if in some representation of S the dimension d is at most κ, the number of equations and inequalities is at most κ, and each of them has degree at most κ. (See [ 13 ].)”
Solving non-linear arithmetic (2012, methodology): “Due to the lack of space and the volume of algorithms and concepts involved, we concentrate on the details of the decision procedure in this paper and refer the reader to the existing literature for further information [8,9,10,27,3].”