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On two problems of Carlitz and their generalizationsстатья
- Автор: Baoulina Ioulia N.
- Сборник: Finite Fields: Theory, Fundamental Properties and Applications
- Серия: Mathematics Research Developments
- Глава в коллективной монографии
- Год издания: 2017
- Место издания: Nova Science Publishers New York
- Первая страница: 147
- Последняя страница: 159
- Аннотация: Let N_q be the number of solutions to the equation (a_1 x_1^{m_1}+...+a_n x_n^{m_n})^k=bx_1^{k_1}...x_n^{k_n} over the finite field F_q=F_{p^s}. Carlitz found formulas for N_q when k_1=...=k_n=m_1=...=m_n=1, k=2, n=3 or 4, p>2; and when m_1=...=m_n=2, k=k_1=...=k_n=1, n=3 or 4, p>2. In earlier papers, we studied the above equation with k_1=...=k_n=1 and obtained some generalizations of Carlitz's results. Recently, Pan, Zhao and Cao considered the case of arbitrary positive integers k_1,...,k_n and proved the formula N_q=q^{n-1}+(-1)^{n-1}, provided that gcd(\sum_{j=1}^n (k_jm_1... m_n/m_j)-km_1... m_n,q-1)=1. In this chapter, we determine N_q explicitly in some other cases.
- Добавил в систему: Баулина Юлия Николаевна