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

$\mathrm{resultant}(h, T) = \mathrm{resultant}(\cdots(\mathrm{resultant}(h, t_s),\ldots, t_i)\cdots)\neq 0$ ,

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.

Read more about this topic: Regular Chain

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