Define a ring of polynomials.
Define a set of polynomials of R. Each of them will be viewed as an equality to 0.
Ideally, you would like to decompose the set of the common solutions of sys into a list of points. The Triangularize command does this by using symbolic expressions. Sometimes several points are grouped together in a generic one, as in this example. These groups of points are called regular chains, and they are grouped together because they share some mathematical properties.
Since regular chains may contain large expressions, their output form is just a word. To view their members, use the Equations command.
The last three regular chains are very simple: each of them clearly corresponds to a single point. The first regular chain corresponds to two points, because its univariate polynomial in the "smallest" variable has two roots.
Consider now another polynomial ring and another polynomial system.
In the polynomial ring, the ordering on the variables is such that . Solving with this ordering implies that you want to express and as functions of the other variables. Hence you can view the system as a parametric linear system with two equations and two unknowns, and . Applying RegularChains[Triangularize] displays the generic solution, which is similar to the solution given by Groebner[Solve].
This generic solution assumes that the determinant of the system is not zero. With the option output=lazard, Triangularize gives all of the solutions, including those that cancel the determinant of the input system.
You already know that each regular chain is associated with a set of equations. It is also associated with a set of inequations.
The inequations of a regular chain are the set of the initials of the polynomials of . In the first regular chain above, the inequations are and . Hence, for this regular chain, none of these two polynomials should vanish. The solutions of that cancel either or the determinant are given by the other regular chains of . Below, for each regular chain of , we print its list of equations together with its set of inequations.
Assume now that you want to see and as transcendental quantities; that is, quantities that cannot satisfy any polynomial equations. Then you need to redefine the polynomial ring as follows.
Now, you can obtain five regular chains, none of them imposing a condition on or . The following example uses the option probability, by which a triangular decomposition is done by using a modular algorithm.
