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