Explicit Hilbert’s irreducibility theorem in function fields
Lior Bary‐Soroker, Alexei Entin · Contemporary mathematics - American Mathematical Society · 2021
We prove a quantitative version of Hilbert’s irreducibility theorem for function fields: If f ( T 1 , … , T n , X ) f(T_1,\ldots , T_n,X) is an irreducible polynomial over the field of rational functions in u u over a finite field with q q elements, then the proportion of n n -tuples ( t 1 , … , t n ) (t_1,\ldots , t_n) of monic polynomials in u u of degree d d for which f ( t 1 , … , t n , X ) f(t_1,\ldots , t_n,X) is reducible out of all n n -tuples of degree d d monic polynomials is O ( d q − d / 2 ) O(dq^{-d/2}) .