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D.8.4.4 mp_res_mat
Procedure from library solve.lib (see solve_lib).
- Usage:
- mp_res_mat(i [, k] ); i ideal, k integer,
k=0: sparse resultant matrix of Gelfand, Kapranov and Zelevinsky,
k=1: resultant matrix of Macaulay (k=0 is default)
- Assume:
- The number of elements in the input system must be the number of
variables in the basering plus one;
if k=1 then i must be homogeneous.
- Return:
- module representing the multipolynomial resultant matrix
Example:
| | LIB "solve.lib";
// compute resultant matrix in ring with parameters (sparse resultant matrix)
ring rsq= (0,u0,u1,u2),(x1,x2),lp;
ideal i= u0+u1*x1+u2*x2,x1^2 + x2^2 - 10,x1^2 + x1*x2 + 2*x2^2 - 16;
module m = mp_res_mat(i);
print(m);
==> -16,0, -10,0, (u0),0, 0, 0, 0, 0,
==> 0, -16,0, -10,(u2),(u0),0, 0, 0, 0,
==> 2, 0, 1, 0, 0, (u2),0, 0, 0, 0,
==> 0, 2, 0, 1, 0, 0, 0, 0, 0, 0,
==> 0, 0, 0, 0, (u1),0, -10,(u0),0, -16,
==> 1, 0, 0, 0, 0, (u1),0, (u2),(u0),0,
==> 0, 1, 0, 0, 0, 0, 1, 0, (u2),2,
==> 1, 0, 1, 0, 0, 0, 0, (u1),0, 0,
==> 0, 1, 0, 1, 0, 0, 0, 0, (u1),1,
==> 0, 0, 0, 0, 0, 0, 1, 0, 0, 1
// computing sparse resultant
det(m);
==> (-2*u0^4+18*u0^2*u1^2+4*u0^2*u1*u2+22*u0^2*u2^2-16*u1^4+80*u1^3*u2-52*u1^\
2*u2^2-120*u1*u2^3-36*u2^4)
// compute resultant matrix (Macaulay resultant matrix)
ring rdq= (0,u0,u1,u2),(x0,x1,x2),lp;
ideal h= homog(imap(rsq,i),x0);
h;
==> h[1]=(u0)*x0+(u1)*x1+(u2)*x2
==> h[2]=-10*x0^2+x1^2+x2^2
==> h[3]=-16*x0^2+x1^2+x1*x2+2*x2^2
module m = mp_res_mat(h,1);
print(m);
==> x0, x1, x2, 0, 0, 0, 0,0, 0, 0,
==> 0, x0, 0, x1,x2,0, 0,0, 0, 0,
==> 0, 0, x0, 0, x1,x2,0,0, 0, 0,
==> -10,0, 0, 1, 0, 1, 0,0, 0, 0,
==> 0, 0, 0, 0, x0,0, 0,x1,x2,0,
==> -16,0, 0, 1, 1, 2, 0,0, 0, 0,
==> 0, -10,0, 0, 0, 0, 1,0, 1, 0,
==> 0, 0, -10,0, 0, 0, 0,1, 0, 1,
==> 0, -16,0, 0, 0, 0, 1,1, 2, 0,
==> 0, 0, -16,0, 0, 0, 0,1, 1, 2
// computing Macaulay resultant (should be the same as above!)
det(m);
==> 2*x0^4-18*x0^2*x1^2-4*x0^2*x1*x2-22*x0^2*x2^2+16*x1^4-80*x1^3*x2+52*x1^2*\
x2^2+120*x1*x2^3+36*x2^4
// compute numerical sparse resultant matrix
setring rsq;
ideal ir= 15+2*x1+5*x2,x1^2 + x2^2 - 10,x1^2 + x1*x2 + 2*x2^2 - 16;
module mn = mp_res_mat(ir);
print(mn);
==> 15,0, -10,0, 0, 0, 0, -16,0, 0,
==> 5, 15,0, -10,0, 0, 0, 0, -16,0,
==> 0, 5, 1, 0, 0, 0, 0, 2, 0, 0,
==> 0, 0, 0, 1, 0, 0, 0, 0, 2, 0,
==> 2, 0, 0, 0, 15,0, -10,0, 0, -16,
==> 0, 2, 0, 0, 5, 15,0, 1, 0, 0,
==> 0, 0, 0, 0, 0, 5, 1, 0, 1, 2,
==> 0, 0, 1, 0, 2, 0, 0, 1, 0, 0,
==> 0, 0, 0, 1, 0, 2, 0, 0, 1, 1,
==> 0, 0, 0, 0, 0, 0, 1, 0, 0, 1
// computing sparse resultant
det(mn);
==> -7056
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