Kernel: SageMath 10.4

In [1]:

#Case g+4: s_1^2, s_2^2, s_1s_2, s_1u + s_1ju, s_2u + s_2ju, s_1u′ + s_1ju′, s_2u′ + s_2ju are linear indep.
reset()

In [1]:

var('l1,l2,l3,l4, l5, l6, l7, s2p, up, jup, vp, jvp, s1q, uq, juq, vq, jvq, s1x, s2x, ux, vx, jux, jvx, s1y, s2y, uy, vy, juy, jvy, s1m, s2m, um, vm, jum, jvn, s1n, s2n, un, vn, jun, jvn, s1t, s2t, ut, jut, vt, jvt')
Eqs=[l2*s2p^2+l5*(s2p*up+s2p*jup)+l7*(s2p*vp+s2p*jvp)==0, 
l1*s1q^2+l4*(s1q*uq+s1q*juq)+l6*(s1q*vq+s1q*jvq)==0, 
l1*s1x^2+l2*s2x^2+l3*s1x*s2x+l6*(s1x*vx+s1x*jvx)+l7*(s2x*vx+s2x*jvx)==0, 
l1*s1y^2+l2*s2y^2+l3*s1y*s2y+l6*(s1y*vy+s1y*jvy)+l7*(s1y*vy+s1y*jvy)==0, 
l1*s1m^2+l2*s2m^2+l3*s1m*s2m+l4*(s1m*um+s1m*jum)+l5*(s2m*um+s2m*jum)==0,
l1*s1n^2+l2*s2n^2+l3*s1n*s2n+l4*(s1n*un+s1n*jun)+l5*(s2n*un+s2n*jun)==0, l1*s1t^2+l2*s2t^2+ l3*s1t*s2t+ l4*(s1t*ut+s1t*jut)+l5*(s2t*ut+s2t*jut)+l6*(s1t*vt+s1t*jvt)+l5*(s2t*vt+s2t*jvt)==0  ]

In [2]:

unks = [l1,l2,l3,l4, l5, l6, l7]
m = matrix([[lhs.coefficient(u) for u in unks] for eq in Eqs for lhs in [eq.lhs()]])
v = vector([eq.rhs() for eq in Eqs])

In [3]:

[                                  0                               s2p^2                                   0                                   0                    jup*s2p + s2p*up                                   0                    jvp*s2p + s2p*vp]
[                              s1q^2                                   0                                   0                    juq*s1q + s1q*uq                                   0                    jvq*s1q + s1q*vq                                   0]
[                              s1x^2                               s2x^2                             s1x*s2x                                   0                                   0                    jvx*s1x + s1x*vx                    jvx*s2x + s2x*vx]
[                              s1y^2                               s2y^2                             s1y*s2y                                   0                                   0                    jvy*s1y + s1y*vy                    jvy*s1y + s1y*vy]
[                              s1m^2                               s2m^2                             s1m*s2m                    jum*s1m + s1m*um                    jum*s2m + s2m*um                                   0                                   0]
[                              s1n^2                               s2n^2                             s1n*s2n                    jun*s1n + s1n*un                    jun*s2n + s2n*un                                   0                                   0]
[                              s1t^2                               s2t^2                             s1t*s2t                    jut*s1t + s1t*ut jut*s2t + jvt*s2t + s2t*ut + s2t*vt                    jvt*s1t + s1t*vt                                   0]

In [4]:

(0, 0, 0, 0, 0, 0, 0)

In [5]:

m.rank()

7

In [7]:

#Case g=3. s_1^2, s_2^2, s_1s_2 and s1u+s1ju, s2u+s2ju are linear indep
reset()

In [8]:

var('l1,l2,l3,l4, l5, s2p, up, jup, vp, jvp, s1q, uq, juq, vq, jvq, s1x, s2x, ux, vx, jux, jvx, s1y, s2y, uy, vy, juy, jvy, s1m, s2m, um, vm, jum, jvn, s1n, s2n, un, vn, jun, jvn, s1t, s2t, ut, jut, vt, jvt')
Eqs=[l2*s2p^2+l5*(s2p*up+s2p*jup)==0, 
l1*s1q^2+l4*(s1q*uq+s1q*juq)==0, 
l1*s1x^2+l2*s2x^2+l3*s1x*s2x==0, 
l1*s1y^2+l2*s2y^2+l3*s1y*s2y==0, l1*s1t^2+l2*s2t^2+ l3*s1t*s2t+ l4*(s1t*ut+s1t*jut)+l5*(s2t*ut+s2t*jut)==0 ]

In [9]:

unks = [l1,l2,l3,l4, l5]
m = matrix([[lhs.coefficient(u) for u in unks] for eq in Eqs for lhs in [eq.lhs()]])
v = vector([eq.rhs() for eq in Eqs])

In [10]:

[               0            s2p^2                0                0 jup*s2p + s2p*up]
[           s1q^2                0                0 juq*s1q + s1q*uq                0]
[           s1x^2            s2x^2          s1x*s2x                0                0]
[           s1y^2            s2y^2          s1y*s2y                0                0]
[           s1t^2            s2t^2          s1t*s2t jut*s1t + s1t*ut jut*s2t + s2t*ut]

In [11]:

(0, 0, 0, 0, 0)

In [12]:

m.rank()

5

In [13]:

S=solve(Eqs, l1,l2,l3,l4, l5)
S

[[l1 == 0, l2 == 0, l3 == 0, l4 == 0, l5 == 0]]