On zeros of diagonal forms over p-adic fields
نویسندگان
چکیده
منابع مشابه
Simultaneous diagonal equations over p-adic fields
with coefficients aij in O. Write the degree as k = pm with p m. A solution x = (x1, . . . , xN) ∈ K is called non-trivial if at least one xj is non-zero. It is a special case of a conjecture of Emil Artin that (∗) has a non-trivial solution whenever N > Rk. This conjecture has been verified by Davenport and Lewis for a single diagonal equation over Qp and for a pair of equations of odd degree ...
متن کاملZeros of p-adic forms
A variant of Brauer’s induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.
متن کاملDiagonal equations of different degrees over p-adic fields
We study systems of two homogeneous diagonal polynomials of different degrees, giving a bound on the number of variables required to guarantee that the system has a nontrivial Qp-integral zero regardless of the coefficients. We then specialize to the case in which at least one of the degrees is odd and use a different method to show that better bounds may be obtained in this situation.
متن کاملOn p-adic zeros of systems of diagonal forms restricted by a congruence condition
This paper is concerned with non-trivial solvability in p-adic integers of systems of additive forms. Assuming that the congruence equation ax + by + cz ≡ d (mod p) has a solution with xyz 6≡ 0 (mod p) we have proved that any system of R additive forms of degree k with at least 2 · 3R−1 · k + 1 variables, has always non-trivial p-adic solutions, provided p k. The assumption of the solubility of...
متن کاملSystems of Diagonal Equations over J-adic Fields
with coefficients in +. It is an interesting problem in number theory to determine when such a system possesses a nontrivial +-rational solution. In particular, we define Γ*(k,R,+) to be the smallest natural number such that any system of R equations of degree k in N variables with coefficients in + has a nontrivial +-rational solution provided only that N&Γ*(k,R,+). For example, when k ̄ 1, ord...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1987
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-48-3-261-273