depends on the projection phase.

Cylindrical Algebraic Decomposition Algorithm
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CAD takes as input, a set of polynomials in $n$ variables and generates a cylindrical algebraic decomposition of the $n$-dimensional real space. This algorithm works recursively to produce a series of projections on lesser dimensions and builds the CAD of each dimension while returning the recurring functions. An outline of CAD algorithm [@20yrs] is presented below. The intricate details have been omitted as the purpose of this presentation is to provide a glimpse into the algorithm, rather than to study the algorithm itself.

1.  Project each polynomial onto lesser dimensions. Polynomials in $n$ variables are taken as input and a set of polynomials in $n-1$ variables is obtained as output, upon applying a projection operator. This process is continued till a set of univariate polynomials is obtained.

2.  These univariate polynomials can easily be solved (using Sturm’s theorem iteratively or otherwise). The roots of these univariate polynomials and the intervals between them are taken as the regions in the CAD of one dimensional real space. Designate a point in each region as a sample point.

3.  For the two dimensional CAD, substitute the sample point found in each one-dimensional region, 