Algebra`AlgebraicInequalities`

The package provides a function for solving systems of strong polynomial inequalities in one or more unknowns. To be precise, SemialgebraicComponents[ineqs, vars] gives a finite set of solutions of the system of inequalities. That is, within the set of solutions, any solution can be connected by a continuous path to a solution in the finite set. The variable ineqs is a list of strong inequalities, where both sides of each inequality are polynomials in variables vars with rational coefficients. In other words, SemialgebraicComponents[ineqs, vars] gives at least one point in each connected component of the open semialgebraic set defined by inequalities ineqs.

Solutions of inequalities.

This loads the package.

In[1]:= <<Algebra`AlgebraicInequalities`

Here is a point from each of the three intervals forming the set of solutions of .

In[2]:= SemialgebraicComponents[{x (x^2 - 2) (x^2 - 3) > 0}, x]

Out[2]=

This gives one point in each of the two connected components of the set bounded by the circle and the hyperbola .

In[3]:= SemialgebraicComponents[{x^2 + y^2 < 4, x y > 1}, {x, y}]

Out[3]=

This proves that the ball is contained in the ellipsoid .

In[4]:= SemialgebraicComponents[{x^2 + y^2/4 + z^2/9 > 1,
x^2 + (y - 1)^2 + (z - 2)^2 < 1/9},
{x, y, z}]

Out[4]=