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