This type represents a triangular system.

A polynomial system is a pair *(F,H)*, where *F⊂k[x]* is a list of equations and and *H⊂k[x]* is a list of inequations. The *zero set* of the system is

A triangular system *(T,U)* is a polynomial system that satisfies some additional properties [Wang’01]. The most basic property, is that the set *T = {t _{1},...,t_{m}}* must be

Triangular systems have very nice algorithmic properties. In particular, the zero set of a triangular system *(F,H)* is always nonempty, and moreover, has dimension *n-m*. In addition, membership to the saturated ideal can be efficiently decided.

Any polynomial system can be decomposed into triangular systems. The method triangularize can be used to compute such decomposition.

The constructor of this type is triaSystem.

- triangularize -- triangular decomposition of polynomial systems

- RingMap TriaSystem -- apply ring map to a triangular system
- triaSystem(Ring,List), see triaSystem -- a triangular system
- triaSystem(Ring,List,List), see triaSystem -- a triangular system

- degree(TriaSystem) -- product of the degrees in a triangular set
- codim(TriaSystem), see dim(TriaSystem) -- dimension of a triangular set
- dim(TriaSystem) -- dimension of a triangular set
- freeVariables(TriaSystem), see freeVariables -- free variables of a triangular set
- generators(TriaSystem) -- equations of a triangular system
- ineqs(TriaSystem), see ineqs -- inequations of a triangular system
- initial(TriaSystem), see initial -- initial of a polynomial
- isPrimeSimple(TriaSystem), see isPrimeSimple -- simple primality test of triangular systems
- isRegularChain(TriaSystem), see isRegularChain -- whether a triangular set is a regular chain
- isStronglyNormalized(TriaSystem), see isStronglyNormalized -- whether a triangular set is strongly normalized
- resultant(RingElement,TriaSystem) -- iterated resultant by a triangular set
- pseudoRemainder(RingElement,TriaSystem), see RingElement % TriaSystem -- pseudo-remainder by a triangular set
- RingElement % TriaSystem -- pseudo-remainder by a triangular set
- saturate(TriaSystem) -- saturated ideal of a triangular system
- net(TriaSystem)

The object TriaSystem is a type, with ancestor classes HashTable < Thing.