A problem on Diophantine Analysis.

[mathbalarka]

I was toying around recently with diophantine forms \(\displaystyle kX^2 + lY^3 + mZ^5 = 0\) over \(\displaystyle \mathbb{Q}^3\).

After a complex bit of elliptic modular methods applied at it, I managed to (partially) solve a special case (k, l, m) = (1, 1, -1728). The parameterization I found is a triple of 11, 20 and 30 degree homogeneous polynomials (\(\displaystyle u, v \in \mathbb{Q}\)):

\(\displaystyle X = u^{30} + v^{30} + 522(u^{25}v^5 - u^5v^{25}) - 10005(u^{20}v^{10} + u^{10}v^{20}) \\ Y = -u^{20} - v^{20} + 228(u^{15}v^5 - u^5v^{15}) - 494u^{10}v^{10} \\ Z = uv(u^{10} + 11 u^5 v^5 - v^{10}) \)

Indeed, I realized just a few hours ago, that this is the Icosahedral identity and the polynomials are, indeed the icoshardal invariants in homogeneous form. These are, of course, no coincidence which I understood after taking a look at my 9-page calculations.

My question is whether this particular parameterization is the general solution to the diophantine form of interest. I cannot seem to be able to extract that particular property from my calculations, even if it represents many interesting facts.

Thanks in advanced,
Balarka
.

 

[jvicens]

Hello Balarka,

Thank you for sharing your findings on the diophantine form kX^2 + lY^3 + mZ^5 = 0 over \mathbb{Q}^3. It is impressive that you were able to partially solve the special case (k, l, m) = (1, 1, -1728) using elliptic modular methods. Your parameterization of X, Y, and Z as triple of 11, 20 and 30 degree homogeneous polynomials (u, v \in \mathbb{Q}) is indeed interesting.

However, I would caution against assuming that this particular parameterization is the general solution to the diophantine form. It is possible that it may only be a solution for this specific special case, and not for all values of k, l, and m. It would be beneficial to continue exploring and testing your parameterization for different values of k, l, and m to see if it holds true.

I am also curious about the connection you mentioned between your parameterization and the Icosahedral identity and icoshardal invariants. It would be helpful if you could provide more information or references on these concepts for better understanding.

Overall, your findings are definitely interesting and worth further investigation. Keep up the good work!