i1 : kk = ZZ/101; |
i2 : S = kk[a..f]; |
i3 : I = minors(2, genericSymmetricMatrix(S, 3)) 2 2 o3 = ideal (- b + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c + a*f, - ------------------------------------------------------------------------ 2 c*e + b*f, - c*d + b*e, - c*e + b*f, - e + d*f) o3 : Ideal of S |
i4 : pts = randomPointsOnRationalVariety(I, 4) o4 = {| -25 20 -30 -16 24 -36 |, | 19 -29 19 23 -29 19 |, | -44 46 -8 7 -10 ------------------------------------------------------------------------ -29 |, | 8 41 -24 46 -22 -29 |} o4 : List |
i5 : for p in pts list sub(I, p) == 0 o5 = {true, true, true, true} o5 : List |
i6 : S = kk[a..d]; |
i7 : F = groebnerFamily ideal"a2,ab,ac,b2" 2 2 2 o7 = ideal (a + t b*c + t a*d + t c + t b*d + t c*d + t d , a*b + t b*c + 1 3 2 4 5 6 7 ------------------------------------------------------------------------ 2 2 2 t a*d + t c + t b*d + t c*d + t d , a*c + t b*c + t a*d + t c + 9 8 10 11 12 13 15 14 ------------------------------------------------------------------------ 2 2 2 t b*d + t c*d + t d , b + t b*c + t a*d + t c + t b*d + t c*d 16 17 18 19 21 20 22 23 ------------------------------------------------------------------------ 2 + t d ) 24 o7 : Ideal of kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ][a..d] 6 5 12 2 4 11 18 24 1 3 8 10 17 23 7 9 14 16 20 22 13 15 19 21 |
i8 : J = groebnerStratum F; o8 : Ideal of kk[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ] 6 5 12 2 4 11 18 24 1 3 8 10 17 23 7 9 14 16 20 22 13 15 19 21 |
i9 : compsJ = decompose J; |
i10 : compsJ = compsJ/trim; |
i11 : #compsJ == 2 o11 = true |
i12 : compsJ/dim o12 = {11, 8} o12 : List |
There are 2 components. We attempt to find points on each of these two components. We are successful. This indicates that the corresponding varieties are both rational. Also, if we can find one point, we can find as many as we want.
i13 : netList randomPointsOnRationalVariety(compsJ_0, 10) +-----------------------------------------------------------------------------------------+ o13 = || 42 -50 -50 38 -39 6 -1 47 49 -18 18 -28 -47 19 48 34 -13 11 -16 -38 39 21 -43 -39 | | +-----------------------------------------------------------------------------------------+ || -44 -31 -9 -21 -42 -36 -47 -20 38 -34 -3 -43 22 16 -35 2 -48 32 -28 -15 -47 38 -47 45 || +-----------------------------------------------------------------------------------------+ || 45 -29 -1 -4 42 -35 4 -13 18 -17 1 21 39 -23 50 15 -11 -11 19 47 -16 7 48 43 | | +-----------------------------------------------------------------------------------------+ || 35 -44 -33 -8 21 -2 -44 -20 19 -28 19 27 11 40 34 33 1 -14 35 36 11 -38 -3 46 | | +-----------------------------------------------------------------------------------------+ || 33 47 46 16 -22 -25 -44 -36 -30 -37 30 -25 -47 29 -41 2 -13 -41 -47 22 -23 -7 -10 15 | | +-----------------------------------------------------------------------------------------+ || 43 1 -41 23 -42 -14 37 -50 -32 -20 -5 -49 -9 32 -18 -22 24 43 -18 30 39 27 -30 -32 | | +-----------------------------------------------------------------------------------------+ || 12 8 26 15 22 12 0 -5 6 17 -21 -18 -33 -49 -19 33 -20 0 -15 -48 39 0 44 -19 | | +-----------------------------------------------------------------------------------------+ || -48 -27 -8 -33 -35 -16 -31 -44 -46 -49 -21 -3 -26 13 -40 4 -11 -48 36 -39 9 -39 -8 22 || +-----------------------------------------------------------------------------------------+ || 34 29 30 -1 34 -22 32 -6 39 -28 -15 37 41 -30 47 -22 -6 -7 -8 43 36 -3 35 16 | | +-----------------------------------------------------------------------------------------+ || 10 -5 -38 -25 -21 21 14 42 19 -41 8 -35 25 -31 29 3 -49 38 -35 -9 6 40 -13 -2 | | +-----------------------------------------------------------------------------------------+ |
i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) +----------------------------------------------------------------------------------------+ o14 = || 1 4 44 1 40 20 29 -26 42 -16 -4 50 -40 48 23 -34 -35 37 30 4 -47 27 -31 0 | | +----------------------------------------------------------------------------------------+ || -29 -11 2 28 -12 -21 -30 20 -1 -42 27 -45 -48 -14 -3 -45 -37 30 -31 -39 -48 -29 47 0 || +----------------------------------------------------------------------------------------+ || -20 14 48 3 14 33 18 -46 24 45 -1 -29 1 31 -44 -2 -22 40 28 -49 -18 46 10 0 | | +----------------------------------------------------------------------------------------+ || 21 25 -23 -45 -48 -40 8 -7 -6 -1 14 3 -13 -44 -44 21 -41 3 30 7 13 -17 8 0 | | +----------------------------------------------------------------------------------------+ || -15 26 -17 12 0 -5 35 -11 0 -31 -39 40 49 -21 -42 28 42 -18 -29 8 30 -46 23 0 | | +----------------------------------------------------------------------------------------+ || -17 38 -49 -9 2 -1 -9 2 3 49 33 49 -46 25 1 0 -18 12 15 -28 18 -16 27 0 | | +----------------------------------------------------------------------------------------+ || -50 3 -12 -21 -10 4 36 -14 19 -9 43 -48 44 -35 16 6 20 -39 23 -21 -37 -23 19 0 | | +----------------------------------------------------------------------------------------+ || 36 -17 -27 37 -4 12 42 -16 -15 12 -3 -14 -28 38 -24 18 -9 6 -47 0 -28 47 -33 0 | | +----------------------------------------------------------------------------------------+ || 43 44 31 14 -9 -4 -4 -40 -48 45 47 17 -37 29 -9 10 -28 -33 -29 28 26 5 42 0 | | +----------------------------------------------------------------------------------------+ || 17 -47 -23 8 -18 -45 -25 2 44 41 -19 -37 5 16 2 38 -13 -20 30 44 4 22 -29 0 | | +----------------------------------------------------------------------------------------+ |
This routine expects the input to represent an irreducible variety