Motivation/example. Consider $K = mathbb{Q}(sqrt[3]{2})$. This is a number field with ring of integers $O_K = mathbb{Z}[sqrt[3]{2}]$. We have a norm map $N_{K/mathbb{Q}}$ which maps $x + ysqrt[3]{2} + zsqrt[3]{4}$ to $x^3 + 2y^3 + 4z^6 - 6xyz$; restricting to $mathcal{O}_K$ gives of course the same form. Using standard results about factorization of prime ideals, it is not too hard to see which integers are norms of elements in $mathcal{O}_K$. Therefore we can get the values of $n$ (with some work...) for which $x^3 + 2y^3 + 4z^3 - 6xyz = n$ has integral solutions, and I guess it is also possible to say something about the solutions for a fixed $n$ - although it is not obvious to me how to do this in general.
The same is of course true for many interesting quadratic forms - to cite a famous example: we can get all positive integers $n$ which can be written as the sum of two perfect squares, or more generally as $x^2 + alpha y^2$ (for some interesting values of $alpha$).
Questions. Is this a fruitful method to study diophantine equations? Are there interesting "large" classes of higher degree polynomials/diophantine equations which can be treated by this sort of argument? What is known in general about such "norm forms"? How to decide whether a polynomial is a norm form? Et cetera :)
(I know that it is a bit vague... I didn't find any useful references.)
No comments:
Post a Comment