1
$\begingroup$

Let $w_1,\cdots,w_n$ be of elements of $\Omega$, that is the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by the $-\deg$ valuation.

Assume that $w_1,\cdots,w_n$ are $\mathbb F_q\left(\left(\frac1T\right)\right)$-linearly independent. Can one bound the number of elements of $\mathbb F_q[T] w_1\oplus\mathbb F_q[T] w_2\oplus\cdots\oplus\mathbb F_q[T] w_n$ with degree less than $r$ ($r\in\mathbb R$)?

Improve this question
$\endgroup$
1
  • 3
    $\begingroup$ One can define the successive minima in this setting and give estimates in terms of the successive minima that are similar to those in the integer case, but simpler. Or one can express such a tuple and a real $r$ as a vector bundle on $\mathbb P^1_{\mathbb F_q}$ and express things in terms of the classification of vector bundles on $\mathbb P^1$. The two perspectives are equivalent and it's mainly a matter of taste. $\endgroup$
    – Will Sawin
    Commented Oct 30, 2023 at 1:28

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .