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Definitions of Gauss and Ramanujan sums over the algebras A and Lv are given, and their main properties are proved. Using these results an analogous of an old result of Libri on the number of solutions of algebraic equations with integral coecients modulo a prime power is obtained, and then used to compute the number of solutions of some equations with coecients in Lv. Finally, an analogous of a problem of Nageswara Rao on algebraic equations subject to partitions is solved for equationswith coecients in Lv.

Tipo de documento: Artículo - Article

Palabras clave: Gauss sums, Ramanujan sums, number of solutions of algebraic equations over





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Bol.
Mat.
17(2), 165–192 (2010) 165 Exponential sums, number of solutions of algebraic equations, and Poincaré series Vı́ctor S.
Albis1 Departamento de Matemáticas Universidad Nacional de Colombia Edilmo Carvajal2 Escuela de Matemáticas Universidad Central de Venezuela, Caracas Definitions of Gauss and Ramanujan sums over the algebras Λ and Lv are given, and their main properties are proved.
Using these results an analogous of an old result of Libri on the number of solutions of algebraic equations with integral coefficients modulo a prime power is obtained, and then used to compute the number of solutions of some equations with coefficients in Lv .
Finally, an analogous of a problem of Nageswara Rao on algebraic equations subject to partitions is solved for equations with coefficients in Lv . Keywords: Gauss sums, Ramanujan sums, number of solutions of algebraic equations over finite algebras, Poincaré series. Se dan definiciones de suma de Gauss y de Ramanujan sobre las álgebras Λ y Lv , y se muestran sus principales propiedades.
Usando estos resultados se demuestra un análogo de un viejo teorema de Libri sobre el númeroo de soluciones de ecuaciones algebraicas con coeficientes enteros, el cual se usa luego para calcular el número de soluciones de algunas ecuaciones algebraicas con coeficientes en Lv .
Finalmente, un análogo de un problema de Nageswara Rao sobre ecuaciones algebraicas sujetas a particiones, se resuelve para ecuaciones con coeficientes en Lv . Palabras claves: Sumas de Gauss, sumas de Ramanujan, número de soluciones de ecuaciones algebricas sobre álgebras finitas, series de Poincaré. MSC: 11T24, 11T55. 1 2 vsalbisg@unal.edu.co ecarvaja@euler.ciens.ucv.ve 166 1 Albis and Carvajal, Exponential sums, number of solutions · · · Introduction In [16] Kummer generalizes Gauss quadratic sums to the rings Z-p` Z, where p is a prime number and ` 0 is a natural number, obtaining some of their most important proper...






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