Q1. Des procédures géométriques 

>	cercle:=proc(pt) x[pt]^2+y[pt]^2-1 end;

 

Cette façon de coder un segment rajoute une variable, mais permet de forcer les trois points à être confondus lorsque deux d'entre eux le sont. 

>	droite:=proc(pt1,pt2,mid) local T; T:=t[pt1,pt2,mid]; T*x[pt1]+(1-T)*x[pt2]-x[mid],T*y[pt1]+(1-T)*y[pt2]-y[mid] end;

 

Q2. Le système et une base de Gröbner 

>	sys:=subs(t[P,Q,M]=1/2,{seq(cercle(pt),pt=[A,B,P,Q,C,D]), seq(droite(op(segment)),segment=[[A,B,M],[P,Q,M],[C,D,M],[B,C,Y],[A,D,X],[P,M,X],[Q,M,Y]]),x[M],y[P]-y[Q],x[P]+x[Q]});

 

>	vars:=op(indets(sys));

 

>	G1:=Groebner[Basis](sys ,tdeg(vars)):

 

LowerCase: Defining mstring_lowercase

 

-> FGb domain: rat_int_cof

 

503 polynomials, 11212 terms total time:        10.576 sec ------------------------------

 

Q3. Appartenance à l'idéal 

>	Groebner[Reduce]((x[C]+x[D]),G1,tdeg(vars));

 

Donc ce n'est pas dans l'idéal. 

>	testrad:=Groebner[Basis]([op(G1),1-t*(x[C]+x[D])],tdeg(vars,t)):

 

-> FGb

 

domain: rat_int_cof

 

481 polynomials, 11303 terms total time:        10.277 sec ------------------------------

 

>	evalb(testrad=[1]);

 

Ni dans le radical. 

Q4. Une base sans les cas dégénérés 

>	varsrat:=op(subs(y[M]=NULL,y[A]=NULL,y[C]=NULL,[vars]));

 

>	infolevel[all]:=2:

 

>	G2:=Groebner[Basis](sys ,tdeg(varsrat)):

 

-> F4 algorithm

 

gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod

 

gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M] y[C]] gcd/LinZip:    degree bounds:  [1 1] gcd/LinZip:    max degrees  :  [3 2] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M] y[A]] gcd/LinZip:    degree bounds:  [1 1] gcd/LinZip:    max degrees  :  [3 2] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [5] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [5] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod

 

gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [5] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [5] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[A] y[M]] gcd/LinZip:    degree bounds:  [1 4] gcd/LinZip:    max degrees  :  [2 8] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[A] y[M]] gcd/LinZip:    degree bounds:  [1 2] gcd/LinZip:    max degrees  :  [2 4] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[A] y[M]] gcd/LinZip:    degree bounds:  [1 2] gcd/LinZip:    max degrees  :  [1 4] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [5] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [6] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [5] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[A] y[M]] gcd/LinZip:    degree bounds:  [1 2] gcd/LinZip:    max degrees  :  [3 8] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[C] y[A] y[M]] gcd/LinZip:    degree bounds:  [1 1 2] gcd/LinZip:    max degrees  :  [2 2 6] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [6] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod

 

gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[C] y[M]] gcd/LinZip:    degree bounds:  [1 1] gcd/LinZip:    max degrees  :  [1 3] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[C] y[M]] gcd/LinZip:    degree bounds:  [1 1] gcd/LinZip:    max degrees  :  [1 4] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M] y[C]] gcd/LinZip:    degree bounds:  [1 1] gcd/LinZip:    max degrees  :  [4 2] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M] y[C]] gcd/LinZip:    degree bounds:  [1 1] gcd/LinZip:    max degrees  :  [3 2] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[A] y[M]] gcd/LinZip:    degree bounds:  [1 2] gcd/LinZip:    max degrees  :  [2 3] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[C] y[M]] gcd/LinZip:    degree bounds:  [1 1] gcd/LinZip:    max degrees  :  [1 3] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[C] y[M]] gcd/LinZip:    degree bounds:  [1 2] gcd/LinZip:    max degrees  :  [2 5] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[A] y[M]] gcd/LinZip:    degree bounds:  [1 2] gcd/LinZip:    max degrees  :  [3 9] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[C] y[M]] gcd/LinZip:    degree bounds:  [1 2] gcd/LinZip:    max degrees  :  [3 9] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [4] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[C] y[M]] gcd/LinZip:    degree bounds:  [1 2] gcd/LinZip:    max degrees  :  [3 8] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [7] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[C] y[A] y[M]] gcd/LinZip:    degree bounds:  [2 2 4] gcd/LinZip:    max degrees  :  [3 3 8] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [6] gcd/LinZip: Using 8-byte integer mod

 

gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[M]] gcd/LinZip:    degree bounds:  [2] gcd/LinZip:    max degrees  :  [8] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Core problem gcd/LinZip:    variables    :  [y[A] y[M]] gcd/LinZip:    degree bounds:  [1 2] gcd/LinZip:    max degrees  :  [2 5] gcd/LinZip: Using 8-byte integer mod gcd/LinZip: Using 8-byte integer mod total time:         0.693 sec

 

>	Groebner[Reduce](x[X]+x[Y],G2,tdeg(varsrat));

 

>

nops(G2);

 

Q5. Les cas dégénérés 

>	infolevel[all]:=0: