2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131855We study the $p$-adic absolute value of the roots of the $L$-functions associated to certain twisted character sums, and additive character sums associated to polynomials $P(x^d)$, when $P$ varies among the space of polynomial of fixed degree $e$ over a finite field of characteristic $p$. For sufficiently large $p$, we determine in both cases generic Newton polygons for these $L$-functions, which is a lower bound for the Newton polygons, and the set of polynomials of degree $e$ for which this generic polygon is attained. In the case of twisted sums, we show that the lower polygon defined in \cite{as1} is tight when $p\equiv 1 [de]$, and that it is the actual Newton polygon for any degree $e$ polynomial.The results in this preprint have been strenghened in arXiv:0706.2340; please look at this new preprintNumber TheoryAlgebraic Geometry14G15, 11T23,11L03text