Investigates the relationship between couple density and q-adic valuation in exponential sums, revealing its significance in analyzing L-functions.
We define the notion of couple density ( D , b ) where D is a non-empty subset of Z m and b a fixed element in { 0 , ⋯ , q − 2 } m ; We determine a minimum in terms of the density of the couple ( D , b ) for the q -adic valuation of the sum S ℓ ( F , b ) with F a Laurent polynomial. And we show that this minimum is a bound for the q -adic valuation of the zeros and poles of the associated L -function.
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Antonine Phigareau (2026) studied this question.