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D.7.2.2 solve
Procedure from library solve.lib (see solve_lib).
- Assume:
the ideal is 0-dimensional;
basering has characteristic 0 and is either complex or without parameters;- Return:
list of solutions of the ideal G, depending on n; one solution is a list of complex numbers in the generated output ring (the new basering).
The result is a list L n = 0: a list of all different solutions (L[i]), n != 0: a list of two elements, L[i][1] contains all different solutions with the same multiplicity L[i][2] the multiplicity L is ordered w.r.t. multiplicity (the smallest first).- Note:
If the problem is not 0-dim. the procedure stops with ERROR, if the ideal G is not a lex. standard basis, it is generated with internal computation (Hilbert driven), if the input-ring (with char 0) has the name "<A>", the lexicographical and complex output-ring has the name "<A>C".
Example:
LIB "solve.lib";
// Find all roots of a multivariate ideal using triangular sets:
int d=4;// with these 3 parameters you may construct
int t=3;// very hard problems for 'solve'
int s=2;
int i;
ring A=0,(x(1..d)),dp;
poly p=-1;
for(i=d;i>0;i--){p=p+x(i)^s;}
ideal I=x(d)^t-x(d)^s+p;
for(i=d-1;i>0;i--){I=x(i)^t-x(i)^s+p,I;}
I;
→ I[1]=x(1)^3+x(2)^2+x(3)^2+x(4)^2-1
→ I[2]=x(2)^3+x(1)^2+x(3)^2+x(4)^2-1
→ I[3]=x(3)^3+x(1)^2+x(2)^2+x(4)^2-1
→ I[4]=x(4)^3+x(1)^2+x(2)^2+x(3)^2-1
// the mutiplicity is
vdim(std(I));
→ 81
list l1=solve(I,6,0);
→ // name of new current ring: AC
// the current ring is
AC;
→ // characteristic : 0 (complex:6 digits, additional 6 digits)
→ // 1 parameter : i
→ // minpoly : (i^2+1)
→ // number of vars : 4
→ // block 1 : ordering lp
→ // : names x(1) x(2) x(3) x(4)
→ // block 2 : ordering C
// you must start with char. 0
setring A;
list l2=solve(I,6,1);
→ // name of current ring: AC
// the number of different solutions is
size(l1);
→ 37
// this is equal to
size(l2[1][1])+size(l2[2][1]);
→ 37
// the number of solutions with multiplicity is
size(l2[1][1])*l2[1][2]+size(l2[2][1])*l2[2][2];
→ 81
// the solutions with multiplicity
l2[2][2];
→ 12
// are
l2[2][1];
→ [1]:
→ [1]:
→ 0
→ [2]:
→ 0
→ [3]:
→ 1
→ [4]:
→ 0
→ [2]:
→ [1]:
→ 0
→ [2]:
→ 1
→ [3]:
→ 0
→ [4]:
→ 0
→ [3]:
→ [1]:
→ 1
→ [2]:
→ 0
→ [3]:
→ 0
→ [4]:
→ 0
→ [4]:
→ [1]:
→ 0
→ [2]:
→ 0
→ [3]:
→ 0
→ [4]:
→ 1
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