#### One-dimensional decomposition

To explain the cylindrical algebraic decomposition method, we first perform a semi-algebraic decomposition of , which is the final step in the projection sequence. Once this is explained, then the multi-dimensional case follows more easily.

Let be a set of univariate polynomials,

 (6.18)

which are used to define some semi-algebraic set in . The polynomials in could come directly from a quantifier-free formula (which could even appear inside of a Tarski sentence, as in (6.9)).

Define a single polynomial as . Suppose that has distinct, real roots, which are sorted in increasing order:

 (6.19)

The one-dimensional semi-algebraic decomposition is given by the following sequence of alternating -cells and 0-cells:

 (6.20)

Any semi-algebraic set that can be expressed using the polynomials in can also be expressed as the union of some of the 0-cells and -cells given in (6.20). This can also be considered as a singular complex (it can even be considered as a simplicial complex, but this does not extend to higher dimensions).

Sample points can be generated for each of the cells as follows. For the unbounded cells and , valid samples are and , respectively. For each finite -cell, , the midpoint produces a sample point. For each 0-cell, , the only choice is to use as the sample point.

Example 6..5 (One-Dimensional Decomposition)   Figure 6.31 shows a semi-algebraic subset of that is defined by two polynomials, and . Here, . Consider the quantifier-free formula

 (6.21)

The semi-algebraic decomposition into five -cells and four 0-cells is shown in Figure 6.32. Each cell is sign invariant. The sample points for the -cells are , , , , and , respectively. The sample points for the 0-cells are 0, , , and , respectively.

A decision problem can be nicely solved using the decomposition. Suppose a Tarski sentence that uses the polynomials in has been given. Here is one possibility:

 (6.22)

The sample points alone are sufficient to determine whether is TRUE or FALSE. Once is attempted, it is discovered that is TRUE. The quantifier-elimination problem cannot yet be considered because more dimensions are needed.

Steven M LaValle 2012-04-20