Regular Chain - Formal Definitions

Formal Definitions

The variables in the polynomial ring

are always sorted as x₁ < ... < x_n. A non-constant polynomial f in can be seen as a univariate polynomial in its greatest variable. The greatest variable in f is called its main variable, denoted by mvar(f). Let u be the main variable of f and write it as

,

where e is the degree of f w.r.t. u and is the leading coefficient of f w.r.t. u. Then the initial of f is and e is its main degree.

• Triangular set

A non-empty subset T of is a triangular set, if the polynomials in T are non-constant and have distinct main variables. Hence, a triangular set is finite, and has cardinality at most n.

• Regular chain

Let T = {t₁, ..., t_s} be a triangular set such that mvar(t₁) < ... < mvar(t_s), be the initial of t_i and h be the product of h_i's. Then T is a regular chain if

,

where each resultant is computed with respect to the main variable of t_i, respectively. This definition is from Yang and Zhang, which is of much algorithmic flavor.

• Quasi-component and saturated ideal of a regular chain

The quasi-component W(T) described by the regular chain T is

, that is,

the set difference of the varieties V(T) and V(h). The attached algebraic object of a regular chain is its saturated ideal

.

A classic result is that the Zariski closure of W(T) equals the variety defined by sat(T), that is,

,

and its dimension is n - |T|, the difference of the number of variables and the number of polynomials in T.

• Triangular decompositions

In general, there are two ways to decompose a polynomial system F. The first one is to decompose lazily, that is, only to represent its generic points in the (Kalkbrener) sense,

.

The second is to describe all zeroes in the Lazard sense,

.

There are various algorithms available for triangular decompositions in either sense.