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 = || -31 -46 41 35 -19 0 -23 -24 -30 -43 13 -5 -16 39 20 21 -13 15 34 19 -39 -47 -38 -18 || +---------------------------------------------------------------------------------------+ || -1 33 -40 29 -45 1 -8 46 3 -47 11 30 -28 -47 21 38 -48 -20 2 16 45 22 -15 -34 | | +---------------------------------------------------------------------------------------+ || -24 5 12 41 28 -34 -1 34 -21 48 -39 -1 19 -16 -12 7 -11 -40 15 -23 43 39 47 -17 | | +---------------------------------------------------------------------------------------+ || 11 -44 20 36 3 -14 -39 -16 31 -3 -3 10 35 11 -39 -38 1 1 33 40 46 11 36 -28 | | +---------------------------------------------------------------------------------------+ || 20 13 42 -22 -5 -25 20 13 -13 -10 -30 -29 -47 -23 -40 -7 -13 2 2 29 15 -47 22 -37 | | +---------------------------------------------------------------------------------------+ || 20 -29 3 25 -31 -15 36 -27 37 -30 3 -23 -18 39 -26 27 24 -15 -22 32 -32 -9 30 -20 | | +---------------------------------------------------------------------------------------+ || 11 -40 -9 -1 11 30 -46 -4 -9 44 21 -14 -15 39 -23 0 -20 21 33 -49 -19 -33 -48 17 | | +---------------------------------------------------------------------------------------+ || -48 -39 24 -3 33 -31 36 12 -10 -8 13 39 36 9 39 -39 -11 34 4 13 22 -26 -39 -49 | | +---------------------------------------------------------------------------------------+ || 25 -38 46 -12 -38 9 -39 5 -11 35 49 -34 -8 36 13 -3 -6 50 -22 -30 16 41 43 -28 | | +---------------------------------------------------------------------------------------+ || 28 -37 41 35 -9 -7 -45 -1 -30 -13 6 -25 -35 6 34 40 -49 -50 3 -31 -2 25 -9 -41 | | +---------------------------------------------------------------------------------------+ |
i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) +-------------------------------------------------------------------------------------+ o14 = || -38 23 -11 16 37 24 -35 14 -42 27 -31 -19 37 -7 -42 -33 -35 -31 -40 -47 27 30 4 0 || +-------------------------------------------------------------------------------------+ || 42 -19 17 -24 -26 -22 -36 22 1 26 22 13 30 -35 44 28 -37 47 -48 -48 -29 -31 -39 0 || +-------------------------------------------------------------------------------------+ || -27 23 -7 16 19 18 47 20 47 50 -46 1 40 11 10 -30 -22 10 1 -18 46 28 -49 0 | | +-------------------------------------------------------------------------------------+ || 3 41 33 16 7 32 6 17 23 -29 -19 -43 3 -5 -23 -48 -41 8 -13 13 -17 30 7 0 | | +-------------------------------------------------------------------------------------+ || 50 -40 28 11 7 44 4 1 -14 5 32 -28 -18 24 6 30 42 23 49 30 -46 -29 8 0 | | +-------------------------------------------------------------------------------------+ || 27 47 -5 20 39 -45 -18 -31 33 34 -29 -3 12 -38 39 17 -18 27 -46 18 -16 15 -28 0 | | +-------------------------------------------------------------------------------------+ || -44 38 26 50 34 -9 31 40 12 -49 2 -21 -39 -33 42 21 20 19 44 -37 -23 23 -21 0 | | +-------------------------------------------------------------------------------------+ || -18 8 -43 -20 12 -30 43 -46 38 40 3 39 6 6 -9 -14 -9 -33 -28 -28 47 -47 0 0 | | +-------------------------------------------------------------------------------------+ || -32 47 -49 8 -50 32 40 -46 -30 -14 -17 25 -33 41 34 15 -28 42 -37 26 5 -29 28 0 | | +-------------------------------------------------------------------------------------+ || -13 -40 -37 29 -41 -17 -37 -28 -49 -14 -9 -38 -20 32 29 49 -13 -29 5 4 22 30 44 0 || +-------------------------------------------------------------------------------------+ |
This routine expects the input to represent an irreducible variety