ABSTRACT
This chapter develops some parts of real algebraic geometry that are useful for geometric constraint systems. Real algebraic geometry adapts the methods and ideas from algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling geometric constraints that are physically meaningful, real algebraic geometry is a core mathematical input for geometric constraint systems. Algebraic geometry is fundamentally the study of sets, called varieties, which arise as the common zeroes of a collection of polynomials. Algebraic geometry works best over the complex numbers, because the geometry of a complex variety is controlled by its defining equations. Because the objects of algebraic geometry have finiteness properties, they may be faithfully represented and manipulated on a computer. Algorithm to decompose a semi-algebraic set into a cell complex of semi-algebraic cells adapted to quantifier elimination.
