Sampling from the Haar Measure
This page contains the code for the experiments in Table 3 for the Haar measure:
//we define a list called powers which gives the monomials of degree 3 in 4 variables.
powers := [[0, 0, 0, 3],
[0, 0, 1, 2],
[0, 0, 2, 1],
[0, 0, 3, 0],
[0, 1, 0, 2],
[0, 1, 1, 1],
[0, 1, 2, 0],
[0, 2, 0, 1],
[0, 2, 1, 0],
[0, 3, 0, 0],
[1, 0, 0, 2],
[1, 0, 1, 1],
[1, 0, 2, 0],
[1, 1, 0, 1],
[1, 1, 1, 0],
[1, 2, 0, 0],
[2, 0, 0, 1],
[2, 0, 1, 0],
[2, 1, 0, 0],
[3, 0, 0, 0]];
//We define the function randomSmoothCubicSurface(N,p,possiblePart) where p is a prime number and N is an integer such that we sample the coefficients from the interval `[0, p^(N+1) - 1]`.
function randomSmoothCubicSurface(N,p)
_<x,y,z,w> := PolynomialRing(Rationals(),4);
P3<x,y,z,w> := ProjectiveSpace(Rationals(),3);
listCoef := [0..p^(N+1)-1];
bool := true;
while bool do
randSurface := 0*x;
for part in powers do
coef := Random(listCoef);
monomial := x^part[1] * y^part[2] * z^part[3] * w^part[4];
randSurface := randSurface + coef * monomial;
end for;
S := Surface(P3,randSurface);
bool := IsSingular(S);
end while;
return randSurface;
end function;
//We use the function howManyLines(surface,p) to compute the number of lines on a fixed cubic surface using Groebner basis techniques.
function howManyLines(surface,p)
f := surface;
QQabcd<a,b,c,d> := PolynomialRing(Rationals(),4);
_<s,t> := PolynomialRing(QQabcd,2);
param := [a*s+b*t,c*s+d*t,s,t];
pt0 := [1,0];
pt1 := [1,1];
ptmin1 := [1,-1];
ptinf := [0,1];
g0 := Evaluate(f,[Evaluate(param[i],pt0) : i in [1..4]]);
g1 := Evaluate(f,[Evaluate(param[i],pt1) : i in [1..4]]);
gmin1 := Evaluate(f,[Evaluate(param[i],ptmin1) : i in [1..4]]);
ginf := Evaluate(f,[Evaluate(param[i],ptinf) : i in [1..4]]);
I := Ideal([g0,g1,gmin1,ginf]);
GB := GroebnerBasis(I);
g := GB[#GB];
K := pAdicField(p,300);
newcoeff := [];
for i in [0..27] do
Append(~newcoeff,K!Rationals()!Coefficient(g,4,i));
end for;
_<x> := PolynomialRing(K);
gK := 0*x;
for i in [1..28] do
gK := gK + newcoeff[i]*x^(i-1);
end for;
try nsol := #Roots(gK);
catch e return -1;
end try;
return nsol;
end function;
// We use lineCounts(N,p,M) to determine the distribution of the line-counts for M cubic surfaces.
function lineCounts(N,p,M)
counts := [0 : j in [-1..27]];
for i in [1..M] do
print i;
f := randomSmoothCubicSurface(N,p);
count := howManyLines(f,p);
counts[count+2] := counts[count+2] + 1;
end for;
return counts;
end function;