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The frequency of occurrence of "locally repeated" values of arithmeticfunctions is a common theme in analytic number theory, for instance in theErd\H{o}s-Mirsky problem on coincidences of the divisor function at consecutiveintegers, the analogous problem for the Euler totient function, and thequantitative conjectures of Erd\H{o}s, Pomerance and Sark\H{o}zy and of Graham,Holt and Pomerance on the frequency of occurrences. In this paper we introducethe corresponding problems in the setting of polynomials over a finite field,and completely solve them in the large finite field limit.

On locally repeated values of arithmetic functions over $\mathbb F_q[T]$