User:Ivan.bannwarth/sandbox

Real dimension
The dimension of a set of real points, typically a semialgebraic set, is the dimension of its Zariski closure. There is equivalent definitions of the real dimension of semialgebraic sets $$S$$ :
 * The real dimension of $$S$$ is the dimension of its Zariski closure.
 * The real dimension of $$S$$ is the maximal integer $$d$$ such that there is an injection of $$[0,1]^d$$ in $$S$$.
 * The real dimension of $$S$$ is the maximal integer $$d$$ such that there is a projection of $$S$$ over a $$d$$-dimensional subspace with a non-empty interior.

For an algebraic set defined over the reals (that is defined by polynomials with real coefficients), it may occur that the dimension of the set of its real points differs from its dimension. For example, the algebraic surface of equation $$x^2+y^2+z^2=0$$ is an algebraic variety of dimension two, which has only one real point (0, 0, 0), and thus the real dimension zero.

The real dimension is more difficult to compute than the algebraic dimension.

For the case of set of real solutions of one polynomial equation, there exists a probabilistic algorithm to compute its real dimension.