Zero-Dimensional Ideal

A zero-dimensional ideal is a proper ideal in a polynomial ring over a field for which the quotient ring has Krull dimension zero. Equivalently,

so the quotient ring is a finite-dimensional vector space over . This is also equivalent to the finiteness of the set of common zeros over the algebraic closure of (Sturmfels 2002, pp. 13-14; Cox 2021).

For every variable , a zero-dimensional ideal contains a nonzero univariate polynomial in . Conversely, these conditions for all the variables imply zero dimensionality. For each variable, a Gröbner basis for any monomial order contains a polynomial whose leading monomial is a positive power of that variable. The monomials not divisible by any leading monomial of the Gröbner basis have equivalence classes forming a finite vector basis of . Multiplication matrices represent multiplication on this vector space and turn the original polynomial equations into eigenvalue problems.

Finiteness of the real solution set alone is insufficient. The ideal in has only the origin as a real zero, while its quotient ring is infinite-dimensional.