Practical Determination of the Dimension of an Algebraic Variety

Abstract

The determination of the dimension of an algebraic variety is a problem that, in principle, can be solved in an algorithmic way. If the variety is projective, this can be made through the computation of the Hilbert polynomial; if it is affine, consider a completion of the affine variety; the completion may have larger dimension than the affine variety, but in this case it may have irreducible components contained in the hyperplane at infinity. This can be checked, and if this happens, a decomposition into irreducible components reduces the problem to the other case. Moreover, if the completion of the affine variety is made with respect to the affine immersion given by a standard basis (also called Gröbner basis) with respect to an homogeneous term ordering, no irreducible component may be contained in the hyperplane at infinity. Hence, a standard basis computation is sufficient to decide the question.