Regular Chain - Properties

Properties

Let T be a regular chain in the polynomial ring R.

• The saturated ideal sat(T) is an unmixed ideal with dimension n − |T|.

• A regular chain holds a strong elimination property in the sense that:

.

• A polynomial p is in sat(T) if and only if p is pseudo-reduced to zero by T, that is,

.

Hence the membership test for sat(T) is algorithmic.

• A polynomial p is a zero-divisor modulo sat(T) if and only if and .

Hence the regularity test for sat(T) is algorithmic.

• Given a prime ideal P, there exists a regular chain C such that P = sat(C).

• If the first element of a regular chain C is an irreducible polynomial and the others are linear in their main variable, then sat(C) is a prime ideal.

• Conversely, if P is a prime ideal, then, after almost all linear changes of variables, there exists a regular chain C of the preceding shape such that P = sat(C).

• A triangular set is a regular chain if and only if it is a Ritt characteristic set of its saturated ideal.