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Thue-Mahler equation

asked 3 years ago

Joao Fede gravatar image

Let FZ[X,Y] be an homogenous polynomial. An equation of the form F(X,Y)=dpa11pann, with p1,,pn are prime numbers and dZ is called a Thue-Mahler equation. Is there an explicit function in sage that can solve this kind of equations even if is a simpler case such as F(x,y) = p^a ?. I have been looking for this with no result.

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It seems not. There is Magma code in the master's thesis Implementation of a Thue-Mahler equation solver by Kyle Hambrook.

rburing gravatar imagerburing ( 3 years ago )

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answered 3 years ago

Max Alekseyev gravatar image

updated 0 years ago

One can rely on the Thue equation solver present in PARI. Here is a sample code for solving the equation f(x,y)=n:

def solvethue(f,n):
    assert f.is_homogeneous()
    return gp.thue(f.subs({f.variables()[1]:1}), n).sage()

R.<x,y> = PolynomialRing(QQ)
print( solvethue(x^3+y^3,1) )
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Asked: 3 years ago

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Last updated: Apr 16 '24