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D.12.7.1 nrRootsProbab

Procedure from library rootsmr.lib (see rootsmr_lib).

int: the number of real roots of the ideal I by a probabilistic algorithm

If I is not a Groebner basis, then a Groebner basis will be computed by using std. If I is already a Groebner basis (i.e. if attrib(I,"isSB"); returns 1) then this Groebner basis will be used, hence it must be one w.r.t. (any) global ordering. This may be useful if the ideal is known to be a Groebner basis or if it can be computed faster by a different method.

If n<10 is given, n is the number of digits being used for constructing a random characteristic polynomial, a bigger n is more safe but slower (default: n=5).
If printlevel>0 the number of complex solutions is displayed (default: printlevel=0).

LIB "rootsmr.lib";
ring r = 0,(x,y,z),lp;
ideal i = (x-1)*(x-2),(y-1)^3*(x-y),(z-1)*(z-2)*(z-3)^2;
nrRootsProbab(i);       //no of real roots (using internally std)
==> 9
i = groebner(i);        //using the hilbert driven GB computation
int pr = printlevel;
printlevel = 2;
==> //ideal has 32 complex solutions, counted with multiplicity
==> *********************************************************************
==> * WARNING: This polynomial was obtained using  pseudorandom numbers.*
==> * If you want to verify the result, please use the command          *
==> *                                                                   *
==> * verify(p,b,i)                                                     *
==> *                                                                   *
==> * where p is the polynomial I returned, b is the monomial basis     *
==> * used, and i the Groebner basis of the ideal                       *
==> *********************************************************************
==> 9
printlevel = pr;
See also: nrRootsDeterm; nrroots; randcharpoly; solve.