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Thue-Mahler equation

asked 2021-06-15 20:42:36 +0200

Joao Fede gravatar image

Let $F \in \mathbb{Z}[X,Y]$ be an homogenous polynomial. An equation of the form $$ F(X,Y) = d p_1^{a_1}\cdots p_n^{a_n},$$ with $p_1, \cdots, p_n$ are prime numbers and $d \in \mathbb{Z}$ 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 ( 2021-06-15 21:29:41 +0200 )edit

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answered 2021-06-21 20:19:41 +0200

Max Alekseyev gravatar image

updated 2021-06-21 20:55:46 +0200

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()
    t = gp.thueinit( f.subs({f.variables()[1]:1}) )
    return gp.thue(t,n)

R.<x,y> = PolynomialRing(QQ)
print( solvethue(x^3+y^3,1) )
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Asked: 2021-06-15 20:42:36 +0200

Seen: 89 times

Last updated: Jun 21