# PolySystem -- a polynomial system

## Description

This type stores a polynomial system, the following methods can be used to access a PolySystem:
• ideal -- the ideal generated by the system
• equations -- the list of polynomials in the system
• ring -- the ring containing the polynomials
• jacobian -- the jacobian of the polynomial map
• parameters -- the list of parameter variables (if any)
Only polynomials are displayed (by net); to see the data stored in a witness set use peek.

## For developers:

Required entries in a PolySystem are
• NumberOfVariables of type ZZ
• NumberOfPolys of type ZZ
• PolyMap of type Matrix, a column matrix over a polynomial ring
• Jacobian of type Matrix, the jacobian of PolyMap
 i1 : CC[x,y] o1 = CC [x..y] 53 o1 : PolynomialRing i2 : S = polySystem {x^2+y^2-6, 2*x^2-y} o2 = S o2 : PolySystem i3 : p = point {{1.0+3*ii,2.3+ii}}; i4 : evaluate(S,p) o4 = | -9.71+10.6ii | | -18.3+11ii | 2 1 o4 : Matrix CC <--- CC 53 53 i5 : evaluate(jacobian S, p) o5 = | 2+6ii 4.6+2ii | | 4+12ii -1 | 2 2 o5 : Matrix CC <--- CC 53 53

Basic service functions:

• polySystem -- construct a polynomial system
• evaluate -- evaluate a polynomial system or matrix at a point

## Methods that use a polynomial system :

• "evaluate(PolySystem,Matrix)" -- see evaluate -- evaluate a polynomial system or matrix at a point
• "evaluateJacobian(PolySystem,Point)" -- see evaluate -- evaluate a polynomial system or matrix at a point
• homogenize(PolySystem,Ring,RingElement) -- homogenize a polynomial system
• "equations(PolySystem)"
• "ideal(PolySystem)"
• "isHomogeneous(PolySystem)"
• "jacobian(PolySystem)"
• "net(PolySystem)"
• "numFunctions(PolySystem)"
• "numParameters(PolySystem)"
• "numVariables(PolySystem)"
• "parameters(PolySystem)"
• "ring(PolySystem)"
• "polySystem(PolySystem)" -- see polySystem -- construct a polynomial system
• "residual(PolySystem,Point)" -- see residual -- residual of a polynomial function at a point
• substitute(PolySystem,Ring) -- substitute a ring in a polynomial system
• texMath(PolySystem) (missing documentation)
• "witnessSet(PolySystem,Matrix,List)" -- see witnessSet -- construct a WitnessSet
• "witnessSet(PolySystem,PolySystem,List)" -- see witnessSet -- construct a WitnessSet

## For the programmer

The object PolySystem is a type, with ancestor classes System < MutableHashTable < HashTable < Thing.