Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N（a1x12 ＋ ···＋anxn2 = bx1 ···xs） denote the number of solutions（x1,...,xn） of the equation a1x12 ＋·&#...Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N（a1x12 ＋ ···＋anxn2 = bx1 ···xs） denote the number of solutions（x1,...,xn） of the equation a1x12 ＋···＋ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F＊q,b ∈ F＊q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.展开更多
Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an ...Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ')) (G: H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.展开更多
基金Supported by the National Natural Science Foundation of China (Grant Nos.1097120510771100)
文摘Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N（a1x12 ＋ ···＋anxn2 = bx1 ···xs） denote the number of solutions（x1,...,xn） of the equation a1x12 ＋···＋ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F＊q,b ∈ F＊q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.
文摘Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ')) (G: H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.
