Distribution of the Minimal Denominator Degree in Function Fields
For a random Laurent series over a finite field, this studies the smallest degree of a polynomial denominator Q for which some P/Q approximates it within a given ultrametric radius. By encoding the approximation condition as the vanishing of a Hankel matrix over the finite field, one derives the exact probability distribution and expected value of this minimal degree, the uniqueness of the monic minimal denominator, and higher-dimensional, P-adic, and Hausdorff-dimension refinements.
On the Minimal Denominator Problem in Function Fields
The minimal denominator problem asks, for a randomly chosen point, the smallest degree of a denominator Q for which some fraction P/Q lies within a fixed radius of the point. This paper transports th…