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# 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

## See also

- WitnessSet -- a witness set

## Methods that use a polynomial system:

- `evaluate(PolySystem,Matrix)` -- see evaluate -- evaluate a polynomial system or matrix at a point
- `evaluateJacobian(PolySystem,AbstractPoint)` -- 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
- substitute(PolySystem,Ring) -- substitute a ring in a polynomial system
- `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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NAGtypes/doc-NAGtypes.m2:396:0.