quadric.ipynb

Kernel: SageMath 10.2

p=5
k=GF(p)
V=VectorSpace(k,4)
P=ProjectiveSpace(k,3)

R.<T0,T1,T2,T3> = PolynomialRing(k)
S.<X> = PolynomialRing(k)

f = T2^2 + T2*T3 + 2*T3^2
#f = (T2 - T3)*(T2 - 2*T3)

ff = f(T0=0,T1=0,T2=X,T3=1)
ff.is_irreducible()

True

Q = T0*T1 - f
Q

T0*T1 - T2^2 - T2*T3 - 2*T3^2

def myeval(Q,v):
    return Q(T0=v[0],T1=v[1],T2=v[2],T3=v[3])

Q = [ p for p in P if myeval(Q,p)==0 ]
len(Q)

26

H = MatrixSpace(k,len(Q),4).matrix([ p for p in Q]).T
H

[1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 1 2 3 4 0 1]
[2 2 3 4 4 1 1 2 2 4 1 3 3 4 4 1 1 2 3 3 1 3 2 4 1 0]
[0 4 2 1 3 0 4 1 3 2 2 1 3 0 4 1 3 2 0 4 1 1 1 1 0 0]
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0]

W = VectorSpace(k,len(Q))
Cperp = W.subspace([ h for h in H ])

def weight(v):
    r = [x for x in v if x != 0]
    return len(r)

T.<Y>=PolynomialRing(ZZ)

## compute the weight enumerator, as an element of ZZ[X]
def WE(D):
    return sum([ Y^weight(c) for c in D ])

WE(Cperp)

104*Y^25 + 520*Y^20 + 1

def countIncidentPoints(v):
    return len([ q for q in Q if v*V(list(q))==0 ])

[ countIncidentPoints(v) for v in V if v != 0 ]

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