ArticlePDF Available

Simultaneous diagonal equations over P-adic fields

Authors:

Abstract

Consider a system of R diagonal forms of degree k in N variables over a P-adic field K. Write k = pτm where p is the residue field characteristic and (m, p) = 1. We prove that this system has a non-trivial P-adic solution if N exceeds (Rk)2τ+5, a bound independent of the field K. We prove the same if N exceeds 4nR2k2 where n = [K : Qp]. Both results improve previously known bounds.
Simultaneous diagonal equations over p-adic
fields
D. Brink, H. Godinho, P. H. A. Rodrigues
February 2008
Abstract. Consider a system of R diagonal forms of degree k in N
variables over a p-adic field K. Write k = p
τ
m where p is the residue
field characteristic and (m, p) = 1. We prove that this system has a
non-trivial p-adic solution if N exceeds (Rk)
2τ+5
, a bound independent
of the field K. We prove the same if N exceeds 4nR
2
k
2
where
n = [K : Q
p
]. Both results improve previously known bounds.
Let K be a finite extension of the field of p-adic numbers Q
p
. Let O be the
ring of integers in K and let p be O’s unique maximal ideal. We say that
K is a p-adic field.
Consider R simultaneous diagonal equations
a
11
X
k
1
+ · · · + a
1N
X
k
N
= 0
.
.
.
.
.
.
.
.
.
a
R1
X
k
1
+ · · · + a
RN
X
k
N
= 0
()
with coefficients a
ij
in O. Write the degree as k = p
τ
m with p - m. A
solution x = (x
1
, . . . , x
N
) K
N
is called non-trivial if at least one x
j
is
non-zero. It is a special case of a conjecture of Emil Artin that () has a
non-trivial solution whenever N > Rk
2
. This conjecture has been verified
by Davenport and Lewis for a single diagonal equation over Q
p
and for a
pair of equations of odd degree over Q
p
(see [3] and [4]), but the general
case remains open.
The main results of the present paper are the following two theorems.
Theorem 1. The system () has a non-trivial solution if the number of
variables N exceeds (Rk)
2τ+5
.
Theorem 2. Let n be the degree of the field extension K/Q
p
. Then ()
has a non-trivial solution if N exceeds 4nR
2
k
2
.
Theorem 1 has the virtue of being independent of K and can be compared
with Skinner [11] where the bound N > k
6τ+4
is given for a single diagonal
equation. Theorem 2 is a natural generalisation of Knapp [7, Theorem 1]
1
and improves Dodson [6, Theorem 1] and Knapp [7, Theorem 3]. See also
Skinner [11] for other references.
Define the integer Γ(R, k) as minimal with the property that any system
() with N > Γ(R, k) has a non-trivial solution over K. Then Theorems
1 and 2 can be restated as Γ(R, k) (Rk)
2τ+5
and Γ(R, k) 4nR
2
k
2
,
respectively. The idea of the proof of the theorems is to first solve () in
the finite residue ring O/p
γ
(for a suitable exponent γ), and then lift this
solution to K via a version of Hensel’s lemma.
A solution x O
N
is called primitive if at least one coordinate x
j
is a
unit in O. Define the integer Φ(R, k, ν) as minimal with the property that
any system () with N > Φ(R, k, ν) has a primitive solution modulo p
ν
.
The Chevalley-Warning theorem (see [2, Lemma 4]) states that any sys-
tem of homogeneous polynomials over a finite field has a non-trivial zero
if the number of variables exceeds the sum of the polynomials’ degrees. In
the special case of systems of diagonal equations, the Chevalley-Warning
theorem gives
Φ(R, k, 1) Rk. (1)
For general moduli a, b 1 one has the relation
Φ(R, k, a + b) + 1 (Φ(R, k, a) + 1) (Φ(R, k, b) + 1) . (2)
This is shown using a well-known ”contraction” argument (see examples in
([4] and [11]). The idea is to construct a primitive solution modulo p
a+b
in
N = (Φ(R, k, a) + 1) · (Φ(R, k, b) + 1) variables as follows: First divide the
left hand side of () into Φ(R, k, a) + 1 subsystems of diagonal forms, each
in Φ(R, k, b) + 1 variables, and solve each system primitively modulo p
b
.
Then multiply each of these solutions by a new variable to form a system
of diagonal forms in Φ(R, k, a) + 1 variables. Since every coefficient is a
multiple of p
b
, to solve this new system primitively modulo p
a+b
is basically
to solve it modulo p
a
. This results in a primitive solution modulo p
a+b
to
() which proves (2).
Let A = (a
ij
) be the coefficient matrix of (). A solution x O
N
is called
non-singular if the matrix (a
ij
x
k
j
) has rank R modulo p, or, equivalently, if
the columns of A corresponding to the indices j with x
j
6≡ 0 (mod p) have
rank R modulo p.
The following strong version of Hensel’s lemma is a natural generalisa-
tion of [5, Lemma 9], from p-adic to p-adic fields. The definition of γ here
is somewhat better than the value 2 + 1 often found in the literature (al-
though Alemu [1] has a result for one equation similar to the lemma below).
Lemma 1. Let e be the ramification index of K over Q
p
and define
γ :=
1 for τ = 0,
e(τ + 1) for τ > 0 and p 6= 2,
e(τ + 2) for τ > 0 and p = 2.
The system () then has a non-trivial solution in K if it has a non-singular
solution modulo p
γ
.
2
Proof. We first show that a unit u O
is a kth power if u ξ
k
(mod p
γ
)
for some ξ O
. This is the standard Hensel’s lemma for τ = 0, so we may
assume τ > 0. Then multiplication x 7→ k·x maps p
e
onto k·p
e
= p
+e
= p
γ
for p 6= 2, and p
2e
onto k · p
2e
= p
γ
for p = 2. For any n > e/(p 1), the p-
adic exponential function and the p-adic logarithm are inverse isomorphisms
between the additive group p
n
and the the multiplicative group 1 + p
n
([9,
Kapitel II, Satz 5.5]). It follows that exponentiation x 7→ x
k
maps 1 + p
e
(for p 6= 2) and 1 + p
2e
(for p = 2) onto 1 + p
γ
. The diagram shows the
situation for p 6= 2:
1 + p
e
x7→x
k
// //
log
1 + p
γ
p
e
x7→k·x
// //
p
γ
exp
OO
Therefore, the elements of the set ξ
k
·(1+ p
γ
) = ξ
k
+p
γ
, to which u belongs,
are all kth powers.
Now let x = (x
1
, . . . , x
N
) be a non-singular solution to () modulo p
γ
.
We may assume x
1
, . . . , x
R
6≡ 0 (mod p) and that the first R columns of
A have rank R modulo p, i.e. form a non-singular matrix modulo p. Row
operations on A will not change the solution set, so we may assume
A =
a
11
0 a
1,R+1
. . . a
1N
.
.
.
.
.
.
.
.
.
0 a
RR
a
R,R+1
. . . a
RN
with a
11
, . . . , a
RR
6≡ 0 (mod p). For each i = 1, . . . , R we have x
k
i
u
i
(mod p
γ
) with u
i
= (a
i,R+1
x
k
R+1
+ · · · + a
iN
x
k
N
)/a
ii
. By the above,
the equation X
k
= u
i
has a solution x
0
i
because it has the solution x
i
mod-
ulo p
γ
. We conclude that (x
0
1
, . . . , x
0
R
, x
R+1
, . . . , x
N
) solves ().
The notion of a p-normalised system of diagonal equations over Q
p
was
introduced in [5]. It is shown there that any system of the form () over
Q
p
has a non-trivial solution provided that any p-normalised system has a
non-trivial solution. All of this is easily generalised to π-normalised systems
with p-adic coefficients (see [7] for details).
Let µ(d) be the maximal number of columns of the coefficient matrix A
which, when considered modulo p, lie in a d-dimensional subspace of F
N
q
.
The key property of π-normalised systems is the inequality
µ(d) N (R d)N/Rk for d = 0, . . . , R 1. (3)
This is [5, Lemma 11] combined with [2, eq. (9)]. An equivalent statement
of this inequality is that any matrix having (R d) rows which are linear
combinations of the rows of A, independent modulo p, contains at least
(R d)N/Rk columns which are nonzero modulo p.
The following slight strengthening of [2, Lemma 2] essentially gives one
extra non-singular submatrix.
3
Lemma 2. Suppose () is π-normalised and has more than k(tR 1)
variables, where t is arbitrary. Then the coefficient matrix A contains t
disjoint R × R submatrices which are non-singular modulo p.
Proof. For every d = 0, . . . , R 1, the assumption N > k(tR 1) combined
with (3) implies µ(d) N (R d)t since µ(d) is integral. Now the
conclusion follows by a combinatorial result of Aigner (see [8, Lemma 1] or
the comment before [2, Lemma 2]).
Next, we extend and improve [2, Lemma 5] using the same idea of proof.
Lemma 3. Suppose () is π-normalised and has more than Rk ·Φ(R, k, ν)
k(R 1)
2
variables, where ν is arbitrary. Then () has a non-singular
solution modulo p
ν
.
Proof. Suppose first () has N = k(tR 1) + 1 variables for some t to be
defined later. Then, by Lemma 2, A has t disjoint R×R submatrices which
are non-singular modulo p. Discard all variables not belonging to one of
these t submatrices. Then we have tR variables left. In each of all but one
of the t submatrices, replace all R variables by one new variable. Then we
have a new system with t 1 + R variables. This system, by definition,
has a primitive solution modulo p
ν
if t 1 + R > Φ(R, k, ν), hence if
t = Φ(R, k, ν) R + 2. Not all the new variables of this solution can be
zero modulo p since the columns corresponding to the old variables form a
non-singular submatrix modulo p and so are linearly independent modulo p.
Therefore, ”inflating” the new variables again gives a non-singular solution
to our original system () in N = Rk · Φ(R, k, ν) k(R 1)
2
+ 1 variables,
and the lemma is proved.
Recall that Γ(R, k) is the minimal integer such that any system () with
N > Γ(R, k) has a non-trivial solution. From Lemmas 1 and 3 it follows
that
Γ(R, k) Rk · Φ(R, k, γ) k(R 1)
2
(4)
since any bound on Γ(R, k) may be proved under the assumption that ()
is π-normalised. For degree k not divisible by p, (4) and (1) give
Γ(R, k) (Rk)
2
k(R 1)
2
, (5)
which extends [2, Theorem 3].
Now, Theorem 2 follows from (4) and the following lemma.
Lemma 4. With γ defined as in Lemma 1, we have
Φ(R, k, γ)
(
p(p 1)
1
nRk for p > 2,
4nRk for p = 2.
Proof. To bound Φ(R, k, γ), we must find a primitive solution modulo p
γ
4
to (). The additive group of the finite residue ring O/p
γ
is equal to the
direct sum of n cyclic subgroups of order p
γ/e
,
O/p
γ
= Zλ
1
· · · Zλ
n
.
This can be seen for example by counting the number of elements of any
given order in both groups and noting that these numbers are the same (see
also [1] for a different proof and a more general statement). Writing each
coefficient a
ij
of () as a Z-linear combination of the λ
i
’s, we see that it
suffices to solve nR congruences
c
i1
X
k
1
+ · · · + c
iN
X
k
N
0 (mod p
γ/e
), i = 1, . . . , nR (6)
with coefficients c
ij
Z. We shall only look for solutions x T
N
where
T = {x Q
p
| x
p
= x} is the set of Teichm¨uller representatives. Since
{x
k
| x T} = {x
(k,p1)
| x T}, we may in (6) replace the exponent
k by (k, p 1). Now, by a theorem of Schanuel [10], the system (6) has a
non-trivial solution x T
N
if N > nR(k, p1)(p
γ/e
1)(p1)
1
. Recalling
k = p
τ
m, we see that (k, p 1) divides m and conclude that Φ(R, k, γ) is
bounded by nR(k, p 1)p
τ+1
(p 1)
1
p(p 1)
1
nRk for p 6= 2, and by
4nRk for p = 2.
The next two lemmas and the final proof of Theorem 1 are much inspired
by the ideas presented in Skinner [11].
Lemma 5. Any a O can be written as
a c
p
τ
0
+ πc
p
τ
1
+ π
2
c
p
τ
2
+ · · · + π
p
τ
1
c
p
τ
p
τ
1
(mod p)
with c
j
O and π being a prime element of O.
Proof. If R O is a set of representatives for O/p, then so is {r
p
τ
| r R},
because the map x 7→ x
p
τ
is a bijection F
q
F
q
. Hence, with suitable
r
n
R, we can write
a =
X
n=0
r
p
τ
n
π
n
=
p
τ
1
X
j=0
π
j
X
i=0
r
p
τ
j+ip
τ
π
ip
τ
p
τ
1
X
j=0
π
j
X
i=0
r
j+ip
τ
π
i
!
p
τ
(mod p),
which proves the lemma.
Lemma 6. Φ(R, k, e) Φ(Rp
τ
, m, e).
Proof. We have to find a primitive solution x O
N
to the R congruences
a
i1
X
k
1
+ · · · + a
iN
X
k
N
0 (mod p), i = 1, . . . , R.
Write each polynomial in this system as a sum of p
τ
polynomials using the
above lemma on each coefficient a = a
ij
. Thus it suffices to find a primitive
solution to Rp
τ
congruences
c
p
τ
i1
X
k
1
+ · · · + c
p
τ
iN
X
k
N
0 (mod p), i = 1, . . . , Rp
τ
.
5
Since
c
p
τ
i1
X
k
1
+ · · · + c
p
τ
iN
X
k
N
(c
i1
X
m
1
+ · · · + c
iN
X
m
N
)
p
τ
(mod p),
it suffices to find a primitive solution to the Rp
τ
congruences
c
i1
X
m
1
+ · · · + c
iN
X
m
N
0 (mod p), i = 1, . . . , Rp
τ
.
Such a solution exists by definition for N > Φ(Rp
τ
, m, e).
We can finally prove Theorem 1. Clearly, Φ(Rp
τ
, m, e) is bounded by
Γ(Rp
τ
, m) which is in turn bounded by (Rk)
2
m(Rp
τ
1)
2
by (5) since
m is not divisible by p. For τ = 0 we already have the bound (5) which is
superior to the one given in Theorem 1. So assume τ > 0. Then Lemma 6
implies
Φ(R, k, e) < (Rk)
2
. (7)
From (4), (2), and (7) it now follows that
Γ(R, k) Rk · Φ(R, k, γ)
Rk · (Φ(R, k, e) + 1)
γ/e
(Rk)
2γ/e+1
(Rk)
2τ+5
.
This concludes the proof of Theorem 1.
References
[1] Y. Alemu, On zeros of diagonal forms over p-adic fields, Acta Arith.
48 (1987), no. 3, 261–273.
[2] J. Br¨udern & H. Godinho, On Artin’s Conjecture, I: Systems of diag-
onal forms, Bull. London Math. Soc. 31 (1998), 305–313.
[3] H. Davenport & D. J. Lewis, Homogeneous additive equations, Proc.
Roy. Soc. London Ser. A 267 (1963), 443–460.
[4] H. Davenport & D. J. Lewis, Two additive equations, Number Theory
(ed. W. J. LeVeque and E. G. Strauss), Proc. Symp. Pure Math. 12
(1969), 74–98.
[5] H. Davenport & D. J. Lewis, Simultaneous equations of additive type,
Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 557–595.
[6] M. Dodson, Some estimates for diagonal equations over p-adic fields,
Acta Arith. 40 (1982), 117–124.
[7] M. P. Knapp, Systems of diagonal equations over p-adic fields, J. Lon-
don Math. Soc. (2) 63 (2001), no. 2, 257–267.
6
[8] L. Low, J. Pitman & A. Wolff, Simultaneous diagonal congruences, J.
Number Theory 29 (1988), 31–59.
[9] J. Neukirch, Algebraische Zahlentheorie, Springer, Berlin, 1992.
[10] S. H. Schanuel, An extension of Chevalley’s theorem to congruences
modulo prime powers, J. Number Theory 6 (1974), 284–290.
[11] C. Skinner, Local solvability of diagonal equations (again), Acta Arith.
124 (2006), no. 1, 73–77.
7
... The proofs of these corollaries, and also the proof of [17,Theorem 10.13], use as input the main result of Skinner [14]. The correction of the latter paper embodied in [15], and improved in [3], provides a substitute for the infelicitous work of [14] that suffices to recover all of these conclusions, with one modification. Namely, the revised version of [16,Corollary 1.3] shows that when d ∈ N and L is a purely imaginary field extension of Q, then v d,r (L) r 2 d−1 e 2 d+1 d (acquiring a factor of 2 in the exponent of e relative to the original statement). ...
Article
Full-text available
We establish that smooth, geometrically integral projective varieties of small degree are not pointless in suitable solvable extensions of their field of definition, provided that this field is algebraic over $\Bbb Q$.
... Os resultados dos dois primeiros capítulos servirão de base para o nosso estudo sobre a existência de soluções para os sistemas de polinômios homogêneos. Finalmente, no capítulo 3, estudaremos os trabalhos desenvolvidos por Alemu [1] e Brink, Godinho e Rodrigues [3] a fim de demonstrarmos os teoremas 1, 3 e 4. ...
Article
Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2009. Faremos um breve estudo sobre corpos locais para obter alguns resultados para corpos p-ádicos. Aproveitando esse estudo, juntamos neste trabalho algumas versões do Lema de Hensel. E baseado nos artigos de Alemu [1] e de Brink, Godinho e Rodrigues [3] veremos algumas condições suficientes sobre o número de variáveis para a solubilidade de sistemas de equações diagonais sobre corpos p-ádicos. _____________________________________________________________________________________ ABSTRACT We will make a brief study of local fields to obtain some results for p-adic fields. Enjoying this study, we join in this work some versions of the Hensels Lemma. Based on the articles of Alemu [1] and Brink, Godinho and Rodrigues [3], we see some sufficient conditions on the number of variables for the solubility of diagonal systems of equations on p-adic fields.
Article
If K is an unramified extension of the p-adic field Qp where p≥3, then we prove that any diagonal homogeneous form defined over K of degree d has a nontrivial zero in K provided that the number of variables is greater than d2.
Article
Full-text available
We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when $\varphi_j\in \mathbb Z[t]$ $(1\le j\le k)$ is a system of polynomials with non-vanishing Wronskian, and $s\le k(k+1)/2$, then for all complex sequences $(\mathfrak a_n)$, and for each $\epsilon>0$, one has \[ \int_{[0,1)^k} \left| \sum_{|n|\le X} {\mathfrak a}_n e(\alpha_1\varphi_1(n)+\ldots +\alpha_k\varphi_k(n)) \right|^{2s} {\rm d}{\boldsymbol \alpha} \ll X^\epsilon \left( \sum_{|n|\le X} |{\mathfrak a}_n|^2\right)^s. \] As a special case of this result, we confirm the main conjecture in Vinogradov's mean value theorem for all exponents $k$, recovering the recent conclusions of the author (for $k=3$) and Bourgain, Demeter and Guth (for $k\ge 4$). In contrast with the $l^2$-decoupling method of the latter authors, we make no use of multilinear Kakeya estimates, and thus our methods are of sufficient flexibility to be applicable in algebraic number fields, and in function fields. We outline such extensions.
Article
The paper contains an investigation of conditions under which R simultaneous equations of additive type in N unknowns have a solution in integers, not all 0. If the degree k of the equations is odd, it suffices if N is greater than an explicit function of R and k. If k is even, two further conditions are imposed, and neither can be entirely avoided. It is also proved that the equations have a solution in p-adic integers, not all 0, if N is greater than an explicit function of R and k.
Article
An investigation into conditions under which an equation of the form c_1x^k_1+ ldots +c_sx^k_s=0 will have solutions in which x_1, ldots, x_s are integers (not all 0).
Article
Let K be a p-adic field, and consider the system F = (F1,…,FR) of diagonal equations (1) with coefficients in K. It is an interesting problem in number theory to determine when such a system possesses a nontrivial K-rational solution. In particular, we define Γ*(k, R, K) to be the smallest natural number such that any system of R equations of degree k in N variables with coefficients in K has a nontrivial K-rational solution provided only that N≥Γ*(k, R, K). For example, when k = 1, ordinary linear algebra tells us that Γ*(1, R, K) = R + 1 for any field K. We also define Γ*(k, R) to be the smallest integer N such that Γ*(k, R, Qp) ≤ N for all primes p.
Article
As a special case of a well-known conjecture of Artin, it is expected that a system of R additive forms of degree k, say [formula] with integer coefficients aij, has a non-trivial solution in Qp for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivial if not all the xi are 0. To date, this has been verified only when R=1, by Davenport and Lewis [4], and for odd k when R=2, by Davenport and Lewis [7]. For larger values of R, and in particular when k is even, more severe conditions on N are required to assure the existence of p-adic solutions of (1) for all primes p. In another important contribution, Davenport and Lewis [6] showed that the conditions [formula] are sufficient. There have been a number of refinements of these results. Schmidt [13] obtained N≫R2k3 log k, and Low, Pitman and Wolff [10] improved the work of Davenport and Lewis by showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, one always encounters a factor k3 log k, in spite of the expected k2 in (2). In this paper we show that one can reach the expected order of magnitude k2. 1991 Mathematics Subject Classification 11D72, 11D79.
Article
Olson determined, for each finite abelian p-group G, the maximal length of a sequence of elements of G such that no subsequence has zero sum, thus settling (at least for these groups) a problem raised by Davenport in connection with factorization in number fields. This problem is equivalent to one on simultaneous linear congruences to which one seeks solutions with the variables restricted to the values 0 and 1. In the present note, the analogous problem for forms of arbitrary degree is settled, again with best possible results. The main tool is an extension of Chevalley's theorem on finite fields to congruences modulo prime powers. This in turn is deduced from Chevalley's theorem by a simple device which circumvents the use of Witt vectors.
Article
Estimates are given for the number of variables required to solve simultaneous diagonal (or additive) congruences, with applications to p-adic equations and equations over GF(p). The main tool is a specialisation of a result on partitioning matroids.
Two additive equations, Number Theory
  • H Davenport
  • D J W J Lewised
  • E G Leveque
  • Strauss
H. Davenport & D. J. Lewis, Two additive equations, Number Theory (ed. W. J. LeVeque and E. G. Strauss), Proc. Symp. Pure Math. 12 (1969), 74–98.