Singular Manual: FamElemsAtEnvCompPoints

| #### D.2.4.18 FamElemsAtEnvCompPoints Procedure from library `grobcov.lib` (see grobcov_lib). Usage: FamElemsAtEnvCompPoints(poly F,ideal C,ideal E); poly F must be the family of hyper-surfaces whose envelop is analyzed. ideal C must be the ideal of restrictions on the variables for defining the family. Must contain less polynomials than the dimension n of the space. ideal E must be a component of the envelop of (F,C), previously computed by envelop. (x_1,..,x_n) are the variables of the hypersurfaces of F, that are considered as parameters of the parametric ring. (u_1,..,u_m) are the parameteres of the hyper-surfaces, that are considered as variables of the parametric ring. Having computed an envelop component E of a family of hyper-surfaces F, with constraints C, it returns the parameter values of the set of all hyper-surfaces of the family passing at one point of the envelop component E. Calling sequence: ring R=(0,x_1,..,x_n),(u_1,..,u_m),lp; poly F=F(x_1,..,x_n,u_1,..,u_m); ideal C=g_1(u_1,..u_m),..,g_s(u_1,..u_m); poly E(x_1,..,x_n); FamElemsAtEnvCompPoints(F,C,E[,options]); Return: list [lpp,basis,segment]. The basis determines the parameter values of the of hyper-surfaces that pass at a fixed point of the envelop component E. The lpp determines the dimension of the set. The segment is the component and is given in Prep. Fixing the values of (x_1,..,x_n) inside E, the basis allows to detemine the values of the parameters (u_1,..u_m), of the hyper-surfaces passing at a point of E. See the not yet published book A. Montes. "Discussing Parametric Polynomial Systems: The Groebner Cover" Note: grobcov is called internally. The basering R, must be of the form Q[a][x] (a=parameters, x=variables). Example:
    LIB "grobcov.lib";
    if(defined(R)){kill R;}
    ring R=(0,y0,x,y),(t),dp;
    short=0;
    poly F=(x-5*t)^2+y^2-3^2*t^2;
    F;
    ==> 16*t^2+(-10*x)*t+(x^2+y^2)
    ideal C;
    C;
    ==> C[1]=0
    def Env=envelop(F,C);
    Env;
    ==> [1]:
    ==>    [1]:
    ==>       _[1]=(3*x-4*y)
    ==>    [2]:
    ==>       [1]:
    ==>          _[1]=1
    ==>    [3]:
    ==>       Normal
    ==>    [4]:
    ==>       1
    ==> [2]:
    ==>    [1]:
    ==>       _[1]=(3*x+4*y)
    ==>    [2]:
    ==>       [1]:
    ==>          _[1]=1
    ==>    [3]:
    ==>       Normal
    ==>    [4]:
    ==>       1
    // E is a component of the envelop:
    ideal E=Env[1][1];
    E;
    ==> E[1]=(3*x-4*y)
    def A=AssocTanToEnv(F,C,E);
    A;
    ==> [1]:
    ==>    [1]:
    ==>       _[1]=t
    ==>    [2]:
    ==>       _[1]=12*t+(-5*y)
    ==>    [3]:
    ==>       [1]:
    ==>          [1]:
    ==>             _[1]=(3*x-4*y)
    ==>          [2]:
    ==>             [1]:
    ==>                _[1]=1
    // The basis of the parameter values of the associated tangent component is
    A[1][2][1];
    ==> 12*t+(-5*y)
    // Thus t=(5/12)*y0 the assocoated tangent family element at (x0,y0) is
    subst(F,t,(5/12)*y0);
    ==> (50*y0^2-75*y0*x+18*x^2+18*y^2)/18
    FamElemsAtEnvCompPoints(F,C,E);
    ==> [1]:
    ==>    [1]:
    ==>       _[1]=t^2
    ==>    [2]:
    ==>       _[1]=144*t^2+(-120*y)*t+(25*y^2)
    ==>    [3]:
    ==>       [1]:
    ==>          [1]:
    ==>             _[1]=(3*x-4*y)
    ==>          [2]:
    ==>             [1]:
    ==>                _[1]=1
    // Thus (12*t^2-5*y0)^2=0 and the unique circle of the family passing at (x0,y0) in E
    // is the associated   tangent circle:
    subst(F,t,(5/12)*y0);
    ==> (50*y0^2-75*y0*x+18*x^2+18*y^2)/18

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