# 120 Quartics

[ ]:

using HomotopyContinuation
using MultivariatePolynomials


We need the function var_coefficients(f,t) to compute the coefficients of a polynomial f with respect to a variable t.

[2]:

#return the t^m-coefficient of f, where t is a variable
function var_coeff(f,t,m)
v = terms(f)
v = filter(a -> (degree(a,t) == m),v)
if length(v) == 0
return polynomial(zeroterm(t))
end
v = map(a -> div(a,1.0*t^m),v)
v = map(a -> subs(a,t => 0),v)
return reduce(+,v)
end

function var_coefficients(f,t)
d = maxdegree(f,t)
v = Array{AbstractPolynomial}(undef,d+1)
for m=0:d
v[d-m+1] = polynomial(var_coeff(f,t,m))
end
return v
end

[2]:

var_coefficients (generic function with 1 method)


We take f to be a quartic in normal form and choose f_2, f_3 as quite generic degree 4 and 6 binary forms.

[3]:

@polyvar z x y  c[0:14] d[0:14]

f = c[1]*x^4 + c[3]*x^3*z + c[4]*x^2*y^2 + c[5]*x^2*y*z + c[6]*x^2*z^2 + c[7]*x*y^3 + c[8]*x*y^2*z + c[9]*x*y*z^2  +  c[11]*y^4 + c[12]*y^3*z + c[13]*y^2*z^2 + c[15]*y*z^3 + c[15]*z^4
p = var_coefficients(f,z)

#use equation (4) on page 18 in Vakils paper
u_2 = p[2]*p[4] - 4p[1]*p[5] - p[3]^2/3
u_3 = p[2]^2*p[5] + p[1]*p[4]^2 - 8p[1]*p[3]*p[5]/3 - p[2]*p[3]*p[4]/3  +2p[3]^3/27

f_2 = x^4 + 3x^3*y + 5x^2*y^2 + 7x*y^3 + 9y^4
f_3 = x^6 + 2*x^5*y + 3*x^4*y^2 + 4*x^3*y^3 + 5*x^2*y^4 + 6*x*y^5 + 7*y^6

[3]:

$$x^{6} + 2x^{5}y + 3x^{4}y^{2} + 4x^{3}y^{3} + 5x^{2}y^{4} + 6xy^{5} + 7y^{6}$$

By demanding the vanishing of the coefficients of u_2 - f_2 and u_3 - f_3 t, we get twelve nonhomogenous equations. HomotopyContinuation.jl solves them for us.

[4]:

h_2 = u_2 - f_2
h_3 = u_3 - f_3
h_2 = subs(h_2,y =>1)
h_3 = subs(h_3,y => 1)

v = var_coefficients(h_2,x)
w = var_coefficients(h_3,x)

s = [v[i] for i in 1:length(v)]
for i in 1:length(w)
push!(s,w[i])
end

F = System(s)
res = solve(F)
sol = solutions(res)

Tracking 234 paths... 100%|█████████████████████████████| Time: 0:00:11
# paths tracked:                  234
# non-singular solutions (real):  120 (8)
# singular endpoints (real):      0 (0)
# total solutions (real):         120 (8)

[4]:

120-element Vector{Vector{ComplexF64}}:
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⋮
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