Triangular decomposition

In computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S1, ..., Se such that a point is a solution of S if and only if it is a solution of one of the systems S1, ..., Se. In computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S1, ..., Se such that a point is a solution of S if and only if it is a solution of one of the systems S1, ..., Se. When the purpose is to describe the solution set of S in the algebraic closure of its coefficient field, those simpler systems are regular chains. If the coefficients of the polynomial systems S1, ..., Se are real numbers, then the real solutions of S can be obtained by a triangular decomposition into regular semi-algebraic systems. In both cases, each of these simpler systems has a triangular shape and remarkable properties, which justifies the terminology. The Characteristic Set Method is the first factorization-free algorithm, which was proposed for decomposing an algebraic variety into equidimensional components. Moreover, the Author, Wen-Tsun Wu, realized an implementation of this method and reported experimental data in his 1987 pioneer article titled 'A zero structure theorem for polynomial equations solving'. To put this work into context, let us recall what was the common idea of an algebraic set decomposition at the time this article was written. Let K be an algebraically closed field and k be a subfield of K. A subset V ⊂ Kn is an (affine) algebraic variety over k if there exists a polynomial set F ⊂ k such that the zero set V(F) ⊂ Kn of F equals V. Recall that V is said irreducible if for all algebraic varieties V1, V2 ⊂ Kn the relation V = V1 ∪ V2 implies either V = V1 or V = V2. A first algebraic variety decomposition result is the famous Lasker–Noether theorem which implies the following. The varieties V1, ..., Ve in the above Theorem are called the irreducible components of V and can be regarded as a natural output for a decomposition algorithm, or, in other words, for an algorithm solving a system of equations in k. In order to lead to a computer program, this algorithm specification should prescribe how irreducible components are represented. Such an encoding is introduced by Joseph Ritt through the following result. We call the set C in Ritt's Theorem a Ritt characteristic set of the ideal ⟨ F ⟩ {displaystyle langle F angle } . Please refer to regular chain for the notion of a triangular set. Joseph Ritt described a method for solving polynomial systems based on polynomial factorization over field extensions and computation of characteristic sets of prime ideals.