Triangular decomposition

In <a href="/facts/Computer_algebra/hjun1KAX">computer algebra</a>, a triangular decomposition of a <a href="/facts/Polynomial_system/1456MrT2">polynomial system</a> 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 <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> of its coefficient <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>, those simpler systems are <a href="/facts/Regular_chain/7rTu8Prq">regular chains</a>. 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 <a href="/facts/Regular_semi-algebraic_system/s81DAK0c">regular semi-algebraic systems</a>. In both cases, each of these simpler systems has a triangular shape and remarkable properties, which justifies the terminology.

Triangular decomposition open-in-new

Triangular decomposition