MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

According to Wikipedia and other sources the Granville-Langevin conjecture states:

If $f$ is a square-free binary form of degree $n > 2$, then for every real $\beta > 2$ there is a constant $C(f,\beta)$ such for all coprime integer $x,y$, the radical of $f(x,y)$ exceeds $C(f,\beta) \max\{|x|,|y|\}^{n-\beta}$

It is equivalent to the $abc$-conjecture.

Is it possible to generalize Granville-Langevin conjecture to number fields or the ring of integers of number fields?

Would someone suggest a generalization?

Such generalization exists for $abc$.

A naive approach is just to take norms.

The relation with $abc$ is that $f(x,y)$ might be a part of a polynomial identity with difference of degrees just $2$. If GL fails, abc will fail because of the small radical.

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.