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# polySystem(GateSystem,PolynomialRing) -- classical polynomial system associated to a gate system

## Synopsis

• Function: polySystem
• Usage:
F = polySystem(G,R)
• Inputs:
• Outputs:
• F, ,

## Description

Given a gate system and a polynomial ring, this function constructs a classical (represented via M2 polynomial map) polynomial system.

 i1 : variables = declareVariable \ {x,y} o1 = {x, y} o1 : List i2 : G = gateSystem(matrix{variables}, matrix{{x*y-1},{x^3+y^2-2},{x^2+2*y-3}}) o2 = gate system: 2 input(s) ---> 3 output(s) o2 : GateSystem i3 : R = CC[X,Y] o3 = R o3 : PolynomialRing i4 : F = polySystem(G,R) o4 = F o4 : PolySystem i5 : evaluate(F,matrix{{1,2}}) o5 = | 1 | | 3 | | 2 | 3 1 o5 : Matrix CC <-- CC 53 53 i6 : evaluate(G,matrix{{1,2}}) o6 = | 1 3 2 | 1 3 o6 : Matrix ZZ <-- ZZ

The ring is expected to be of the form K[x_1..x_n] or K[a_1..a_m][x_1..x_n]. In the latter case, the gate system is expected to take m parameters.

 i7 : variables = declareVariable \ {x,y} o7 = {x, y} o7 : List i8 : params = declareVariable \ {a,b,c} o8 = {a, b, c} o8 : List i9 : G = gateSystem(matrix{params}, matrix{variables}, matrix{{x*y-1},{a*x^2+b*y^2-c}}) o9 = gate system: 2 input(s) ---> 2 output(s) (with 3 parameters) o9 : GateSystem i10 : R = CC[A,B,C][X,Y] o10 = R o10 : PolynomialRing i11 : F = polySystem(G,R) o11 = F o11 : PolySystem i12 : equations F 2 2 o12 = {X*Y - 1, A*X + B*Y - C} o12 : List