J Örg Br

Olivier Robert
email: olivier.robert@univ-st-etienne.fr

RATIONAL POINTS ON LINEAR SLICES OF DIAGONAL HYPERSURFACES

An asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way, we establish two new results on Weyl sums that are of some independent interest.

Introduction

The varieties alluded to in the title line are defined by pairs of equations (1.1)

s j=1 a j x k j = s j=1 b j x j = 0
in which the natural numbers k, s and the integers a j , b j are fixed once and for all. We shall be concerned with deriving an asymptotic formula for the number N (P ) = N a,b (P ) of solutions to (1.1) in integers x j satisfying (1.2) |x j | ≤ P (1 ≤ j ≤ s).

The cases k = 1 and k = 2 are part of the classical theory: When k = 1, the equations (1.1) describe a lattice, and the asymptotic evaluation of N (P ) is elementary. When k = 2, one inserts the linear equation into the quadratic one to eliminate a variable, thus reducing the problem to that of counting those integer points where an integral quadratic form vanishes. For the latter problem, there is a vast literature to which we have nothing to add. Thus, we concentrate on the cases where k ≥ 3.

Theorem 1. Let k ≥ 3, s ≥ 2 k + 2 and suppose that a j = 0 (1 ≤ j ≤ s). Suppose that the pair of equations (1.1) has non-singular solutions in R and in Q p , for all primes p. Then there is a positive number C(a, b) such that

(1.3) N a,b (P ) = C(a, b)P s-k-1 + O P s-k-1 (log P ) -2 .
In algebraic geometry, it is more customary to count rational points on the projective variety defined by (1.1). A rational point on (1.1) corresponds to an integral solution with (x 1 ; x 2 ; . . . ; x s ) = 1. The latter is unique up to sign, and its natural height is defined by max |x j |. By Möbius' inversion formula, the number of rational points on (1.1) with height not exceeding P equals AMS subject classification 11D45 (primary), 11L15, 11P55 (secondary).

1 Subject to the conditions in Theorem 1, this expression is asymptotic to

1 2 ζ(s -k -1) -1 C(a, b)P s-k-1 , as expected.
Theorem 1 should be considered as part of a programme to establish similar asymptotic formulae for intersections of diagonal hypersurfaces, at least when the dimension is suitably large (see [START_REF] Brüdern | On simultaneous diagonal equations and inequalities[END_REF][START_REF] Davenport | Cubic equations of additive type[END_REF][START_REF] Davenport | Simultaneous equations of additive type[END_REF][START_REF] Parsell | Pairs of additive equations of small degree[END_REF][START_REF] Wooley | On simultaneous additive equations[END_REF][START_REF] Wooley | On simultaneous additive equations[END_REF] and the references therein). Recent ground-breaking work of Wooley [START_REF] Wooley | Vinogradov's mean value theorem via efficient congruencing[END_REF][START_REF] Wooley | Vinogradov's mean value theorem via efficient congruencing[END_REF][START_REF] Wooley | The asymptotic formula in Waring's problem[END_REF]] on Vinogradov's mean value theorem has a revolutionary impact in this area. At the time of writing, publicly available descriptions of Wooley's "efficient congruencing" provide conclusions similar to Theorem 1, but subject to a condition slightly milder than s ≥ 2(k -1) 2 when k ≥ 6. However, progress is still ongoing, and Wooley has now announced results that have the potential to supersede Theorem 1 for all k ≥ 5. In the light of this, the main interest is in the cases k = 3 and k = 4, but our proof for k = 4 works equally well for k ≥ 4.

When k = 3, the condition on s in Theorem 1 is s ≥ 10. As in the quadratic case, one may substitute the linear equation into the cubic one to obtain a cubic form in nine or more variables. For cubic forms in nine variables, important work of Hooley [START_REF] Hooley | On nonary cubic forms[END_REF][START_REF] Hooley | On nonary cubic forms[END_REF][START_REF] Hooley | On nonary cubic forms[END_REF] provides an asymptotic formula for the number of its integral zeros within a suitable expanding region, provided that the form is non-singular, a condition that may in some cases be relaxed to allow the singular locus of the form to consist of isolated linearly independent ordinary double points. However, as one readily checks, the projective cubic defined by x j = 0 has 126 singular points, and hence provides an example covered by Theorem 1 but not by Hooley's work. When k = 4, one may again insert the linear equation into the quartic one. This leads to a quartic form that one may analyse by the methods of Birch [START_REF] Birch | Forms in many variables[END_REF] and Browning and Heath-Brown [START_REF] Browning | Rational points on quartic hypersurfaces[END_REF], but this strategy apparently requires s to be as large as 40 or thereabouts, and is therefore is not competitive at present. We prove Theorem 1 by a two-dimensional version of the Hardy-Littlewood method. Our argument rests on a new mean value theorem for the generating function (1.4) f (α, β) = x≤P e(αx k + βx)

that we now describe. Fix a number θ with 2 1-k k < θ ≤ 4 5 and then take Q = P θ . Let m denote the set of real numbers α ∈ [0, 1] for which the inequality |qα -a| ≤ QP -k with q ∈ N, a ∈ Z is only possible when q > Q. due to Vaughan (see [START_REF] Vaughan | On Waring's problem for cubes[END_REF], Theorem B, for k = 3 and [START_REF] Vaughan | On Waring's problem for smaller exponents[END_REF] for k ≥ 4, with a slightly different power of log P ; for refinements see Boklan [START_REF] Boklan | A reduction technique in Waring's problem[END_REF] and Harvey [START_REF] Harvey | Minor arc moments of Weyl sums[END_REF]). Vaughan's approach involves a considerable refinement of the conventional proof of Hua's Lemma (see [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF], Lemma 2.5). Certain divisor sums require treatment by mean value estimates for Hooley's ∆ r -functions to propagate an initial saving through an induction. The initial saving itself comes in through a sieving of the variable of summation in (1.4) and an appeal to a paucity estimate for the diophantine equation

x k 1 + x k 2 = x k 3 + x k 4 .
originally obtained by Hooley [START_REF] Hooley | On the representation of a number as the sum of two h-th powers[END_REF][START_REF] Hooley | On another sieve method and the numbers that are a sum of two hth powers[END_REF][START_REF] Hooley | On another sieve method and the numbers that are a sum of two hth powers[END_REF] (see also Wooley [34], Skinner and Wooley [START_REF] Skinner | Sums of two kth powers[END_REF]). We are able to keep the architecture of Vaughan's treatment largely intact, now building on a paucity estimate for the pair

x k 1 + x k 2 + x k 3 = y k 1 + y k 2 + y k 3 , x 1 + x 2 +
x 3 = y 1 + y 2 + y 3 (Vaughan and Wooley [START_REF] Vaughan | On a certain nonary cubic form and related equations[END_REF] for k = 3, Greaves [START_REF] Greaves | Some Diophantine equations with almost all solutions trivial[END_REF] for k ≥ 4 ; see also de la Bretèche [START_REF] De La Bretèche | Répartition des points rationnels sur la cubique de Segre[END_REF]). Once Theorem 2 is established, it is fairly routine to derive Theorem 1. We postpone a more comprehensive discussion of several complications to the appropriate stage of the argument.

No direct attack on the problem considered here, via the circle method, seems to have been launched in the past, but there is related work of Parsell [START_REF] Parsell | Pairs of additive equations of small degree[END_REF]. Parsell considers more generally a pair of diagonal equations a 1 x k 1 + . . . + a s x k s = b 1 x n 1 + . . . + b s x n s = 0, and applies smooth number technology within a circle method approach to verify the Hasse principle for this pair of equations when s is suitably large. Such a strategy typically supplies a lower bound for the number of solutions within a box which is of the expected order of magnitude. In the special case n = 1 which is the theme of this paper, Parsell proves the Hasse principle for k = 3, s ≥ 10 and for k = 4, s ≥ 17, amongst other results. It is interesting to note that in the case k = 3 his method fails to give a lower bound for N (P ) of the expected size, a defect that is now cured by Theorem 1.

Before we move on to proofs of Theorem 1 and 2, we briefly comment on the condition in Theorem 1 that all a j be non-zero. It suffices to require only that at least 2 k of the a j are non-zero. In fact, the presence of isolated linear variables in (1.1) facilitates the exercise. However, it seems difficult to relax this condition further without improving (1.5), and some lower bound on the number of non-zero a j is definitely necessary. To see this, consider the system

(1.6) 5x 3 1 + 9x 3 2 + 10x 3 3 + 12x 3 4 = x 1 + x 2 + • • • + x 10 = 0.
Here, the cubic equation (when considered in the variables x 1 , . . . , x 4 only) violates the Hasse principle (Cassels and Guy [START_REF] Cassels | On the Hasse principle for cubic surfaces[END_REF]). Thus there are real and p-adic nonsingular solutions of (1.6), but a solution x ∈ Z 10 satisfies x 1 = x 2 = x 3 = x 4 = 0, and it follows easily that there is a positive constant C such that N (P ) = CP 5 + O(P 4 ), in contrast to the leading term of size P s-k-1 in Theorem 1.

Notation. Throughout this paper, small italics a, b, . . . denote integers, and q is a natural number. The letter p is reserved for primes, and k is a natural number with k ≥ 3. Real numbers are denoted by small greek letters α, β, . . .. These conventions apply whenever these symbols do not obviously denote functions. Whenever ε occurs in a statement, it is asserted that the statement is valid for any fixed positive value of ε. Note that if A P ε and B P ε , then we may conclude that AB P ε . The leading parameter is P , and all statements are true whenever P exceeds a certain real number P 0 that depends only on k.

Vectors are in bold face, x = (x 1 , . . . , x s ), the dimension s will depend on the context. To avoid ambiguity, the greatest common factor of a and b is (a; b).

The number of u ∈ N j with u 1 u 2 . . . u j = n is denoted by d j (n), and we write d(n) = d 2 (n) for the number of divisors of n. Similarly, Hooley's functions are defined by

∆ j (n) = max ξ1,...,ξj-1 #{u ∈ N j : u 1 u 2 . . . u j = n, ξ i < log u i ≤ ξ i + 1 (1 ≤ i < j)},
and again we put ∆(n) = ∆ 2 (n). Further, Ω(n) denotes the total number of prime factors of n, counted with multiplicity. Less standard, but common in related literature are the abbreviations

(1.7) L = log P, K = 2 k-1 , J = 2 j-1 .
Acknowledgements. The authors thank the anonymous referee for his very detailed inspection of the manuscript. While this paper was refereed, further progress with the problems considered here was made in the case k = 3, see T.D. Wooley, Mean value estimates for odd cubic Weyl sums, arXiv:1401.7152.

The generating function

Central to the major arc analysis is a good approximation to f (α, β) when α, β are near rational numbers a/q, b/q. The approximating function is built from the expressions For frequent use later on, recall the estimates of Hua

(2.2) S(q, a, b) q (k-1)/k+ε , v(ξ, ζ) P 1 + P k |ξ| + P |ζ| -1/k
that are valid for any real numbers ξ, ζ and natural numbers a, b, q subject to (q; a; b) = 1 (see Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF], Theorems 7.1 and 7.3).

Theorem 3. Let q ∈ N, a, b ∈ Z with (a; q) = 1. Let α, β ∈ R and α = a q + ξ, β = b q + ζ with |ζ| ≤ 1/(2q). Then (2.3) f (α, β) = q -1 S(q, a, b)v(ξ, ζ) + O q (k-1)/k+ε (1 + P k |ξ|) 1/2 . If further |ξ| ≤ 1/(4kqP k-1 ), then (2.4) f (α, β) = q -1 S(q, a, b)v(ξ, ζ) + O q (k-1)/k+ε .
While we still work under the condition that k ≥ 3, it may be worth pointing out that the conclusions in Theorem 3 and their proofs below remain valid when k = 2. However, when k = 2, a stronger version of Theorem 3 (with ε = 0) was obtained recently by Vaughan [START_REF] Vaughan | On generating functions in additive number theory[END_REF]. Apparently, Theorem 3 is new for all k ≥ 3 and the best estimate available hitherto is the special case of [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF], Theorem 7.2, which gives (2.3) with the error term inflated to 1 + P k |ξ| + P |ζ|. It is vital for our later work that both q and |ξ| occur in (2.3) with exponents below 1.

Our proof of Theorem 3 is an adaptation of a standard argument for the classical Weyl sum f (α, 0). We follow Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF], pp. 43-44, quite closely, but differences in detail justify a moderately detailed exposition. By (1.4), (2.1) and orthogonality of additive characters, one confirms the initial identity

(2.5) f (α, β) = 1 q -q/2<r≤q/2 S(q, a, b -r)f ξ, ζ + (r/q) .
We apply a truncated Poisson summation formula to f ξ, ζ + r q . The phase

F (t) = ξt k + ζ + r q t has derivative F (t) = kξt k-1 + ζ + r q
, which is monotonic for t ≥ 0, and for 0 ≤ t ≤ P and |r| ≤ 1 2 q, one has

|F (t)| ≤ k|ξ|P k-1 + 1 2q + 1 2 . With H = [k|ξ|P k-1 ] + 2,
we have now verified the hypotheses of [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF], Lemma 4.2, which gives

f ξ, ζ + r q = |h|≤H v ξ, ζ + r q -h + O(log H).
One uses this within (2.5). On writing r -qh = m and M = q(H + 1 2 ), this produces

f (α, β) = 1 q -M <r≤M S(q, a, b -m)v ξ, ζ + m q + E in which E log H q |r|≤q/2 |S(q, a, b -r)|.
Here, one isolates the term m = 0 and then applies (2.2) to all other terms to conclude that (2.6)

f (α, β) -q -1 S(q, a, b)v(ξ, ζ) q ε-1/k 1≤|m|≤M v ξ, ζ + m q + q (k-1)/k+ε log H.
We proceed to deduce (2.4). In the admissible range for ξ, one has H = 2. We take [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF]. One has F (t) = m q +R with |R| ≤ 3 4q for 0 ≤ t ≤ P , so that F does not change sign and satisfies |F (t)| ≥ |m| 4q . The estimate provided by [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF], Lemma 4.2, then shows that v ξ, ζ + r q q/|m|, and (2.4) is immediate from (2.6).

F (t) = ξt k + ζ + m q t in Lemma 4.2 of Titchmarsh
It remains to prove (2.3) for |ξ| ≥ 1/(4kqP k-1 ), as we now assume. Let F be as before. Its derivative

F (t) = kξt k-1 + ζ + m
q is still monotonic for t ≥ 0, but may have a zero in [0, P ]. We therefore apply a stationary phase argument. For 1 ≤ |m| ≤ M , let T (m) be the set of all t ∈ [0, P ] where |F (t)| ≥ |m| 4q . This is an interval or the union of two intervals, so that [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF], Lemma 4.2, still shows that (2.7)

T (m) e(F (t)) dt q/|m|.
It remains to estimate the contribution from [0, P ] \ T (m). If this is non-empty, then t ∈ [0, P ] \ T (m) satisfies

kξt k-1 + ζ + m q ≤ |m| 4q • This implies that (2.8) 3|m| 4q ≤ |kξt k-1 + ζ| ≤ 5|m| 4q , For 1 ≤ |m| ≤ M , let δ = δ(m) = |ξ| 1/(2k-2) |m|/q (k-2)/(2k-2) .
We claim that (2.9)

[0,P ]\T (m) e(F (t)) dt δ -1 .
To see this, first note that an argument similar to the above shows that the set of all 0 ≤ t ≤ P with δ ≤ |F (t)| ≤ |m| 4q contributes O(δ -1 ). For the remaining t, one has |kξt k-1 + ζ + m q | ≤ δ, and if t 1 , t 2 satisfy this inequality, one finds that k|ξ||t k-1

1 -t k-1 2 | ≤ 2δ. But (2.8) together with |ζ| ≤ 1/(2q) implies that t k-1 j ≥ |m|
4k|ξ|q , so that the binomial expansion yields

|t 1 -t 2 | ≤ 2δ k|ξ| 4k|ξ|q |m| (k-2)/(k-1) 1 δ •
Hence, the set of t ∈ [0, P ] with |F (t)| ≤ δ has measure O(1/δ), and (2.9) follows.

On combining (2.7) and (2.9) to an estimate for v ξ, ζ + m q , we find that

1≤|m|≤M v ξ, ζ + m q q m≤M 1 m + m≤1+2kq|ξ|P k-1 1 δ(m) q 1+ε (1 + P k |ξ|) 1/2 .
In view of (2.6), the desired bound (2.3) is immediate.

We now apply Theorem 3 to establish a strong from of Weyl's inequality for f (α, β). Recall (1.4), (1.7) and the definition of m in Theorem 2.

Lemma 1. Uniformly in β ∈ R, α ∈ m, one has |f (α, β)| K P K-1 L 1+ε .
The proof is essentially that of Lemma 1 in Vaughan [START_REF] Vaughan | On Waring's problem for cubes[END_REF], suitably generalised to k ≥ 3. However, our analysis relies on Theorem 3 rather than [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF]Theorem 4.1], and some care is required to accomodate the weaker error estimates. In particular, it turns out that mimicry of the argument outlined on p. 131 of [START_REF] Vaughan | On Waring's problem for cubes[END_REF] leads to a satisfactory bound only in the case when k ≥ 4, which we temporarily suppose from now on.

Let δ = 1/(100k) and α ∈ m. By Dirichlet's theorem, there are coprime a, q with 1 ≤ q ≤ P k-1-δ and |qα -a| ≤ P 1+δ-k . First suppose that q ≤ P µ where µ = 1 + 1 k+3 . One checks that k-1 k µ < 1 -1 K holds for all k ≥ 4. Hence by Theorem 3 and (2.2), (2.10)

f (α, β) q ε-1/k P 1 + P k α -a q -1/k + P 1-1/K .
Since α ∈ m, we have q > Q or |α -a/q| ≥ Q/(qP k ), and in both cases it follows that

(2.11) f (α, β) Q ε-1/k P + P 1-1/K P 1-1/K , as required.
This leaves the case where q > P µ . Then |α -a/q| ≤ P 1+δ-µ-k , and one checks that

1 + δ -µ = -1 k+3 + δ ≤ -1 K . Consequently, f (α, β) -f a q , β = x≤P e(βx) e(αx k ) -e a q x k P 1-1/K .
Thus, we are reduced to estimating f (a/q, β). We begin by observing that the substitution y = x + h produces

|f (α, β)| 2 = x,y≤P e α(y k -x k ) + β(y -x) = |h|<P e(βh)
1≤x≤P 1≤x+h≤P e α((x + h) k -x k ) (2.12)

≤ P + 0<|h|<P 1≤x≤P 1≤x+h≤P e α((x + h) k -x k ) .
Note that β is absent in this inequality. We now take α = a/q and repeat Weyl differencing in the usual way. Then, as in [START_REF] Vaughan | On Waring's problem for cubes[END_REF], p. 131, one arrives at

|f (a/q, β)| K P K-1 + P K-k 1≤hj ≤P 1≤j≤k-1 min P, k!ah 1 . . . h k-1 q -1
, and one may then complete the estimation in the same way as in the final part of the proof of [26, Lemma 1], but using Hall and Tenenbaum [13, Theorem 70] for ∆ k-1 (n) instead of referencing Hooley [START_REF] Hooley | On a new technique and its applications to the theory of numbers[END_REF]. This completes the proof of Lemma 1 when k ≥ 4. Now suppose that k = 3. More care is required in the part that relies on Weyl differencing. On applying Cauchy's inequality to

f (α, β) = 2 u=1 x≤P x≡u mod 2 e(αx 3 + βx) one finds that |f (α, β)| 2 ≤ 2 x,y≤P x≡y mod 2 e α(x k -y k ) + β(x -y) ,
and the substitution 2z = x + y, 2h = x -y transforms this to

|f (α, β)| 2 ≤ 2 |h|≤P/2 z∈I(h) e(2βh)e 2αh(3z 2 + h 2 )
where I(h) is the subinterval of [1, P ] The double sum over h, z here is that same as the one estimated in Lemma 4 of Vaughan [START_REF] Vaughan | On Waring's problem for cubes[END_REF] with H = P/2, except that in that lemma, 6α is 3α, and the range for z is 1 ≤ z ≤ P . An inspection of the proof shows that these changes are irrelevant, and that the conclusion is still valid in our context. Moreover, similar to an earlier comment, the use of Theorem 70 of Hall and Tenenbaum [START_REF] Hall | Divisors. Cambridge Tracts in Mathematics[END_REF] within the proof of [START_REF] Vaughan | On Waring's problem for cubes[END_REF]Lemma 4] reduces the factor L 4/π+ε in that lemma to L 1+ε . Hence, on choosing 1 ≤ q ≤ P 2-δ and a ∈ Z with |qα -a| ≤ P δ-2 , the augmented form of [START_REF] Vaughan | On Waring's problem for cubes[END_REF]Lemma 4] now shows that |f (α, β)| 4 P 3 L 1+ε holds in all cases where q > P 1+δ . When q ≤ P 1+δ , then Theorem 3 and (2.2) yield (2.10) for k = 3, and one then also confirms (2.11) for k = 3. The proof of Lemma 1 is complete.

We close this section with a technical observation concerning the exponential sum defined in (2.1).

Preparatory mean value estimates

We begin with certain divisor sums that are routinely estimated by van der Corput's method. Let t ∈ N, put x = (x 1 , . . . , x t ) and define the sums

(3.1) Λ(x, y) = t j=1 (x j -y j ), M(x, y) = t j=1 (x k j -y k j ).
Lemma 3. Let t ∈ N, l ∈ N. Then, there exists positive numbers δ and η such that

xj ≤P, yj ≤P M(x,y) =0 d M(x, y) l P 2t L η , xj ≤P, yj ≤P M(x,y) =0 d M(x, y) l e δΩ(M(x,y)) P 2t L η , (3.2) xj ≤P, yj ≤P M(x,y) =0 Λ(x,y)=0 d M(x, y) l P 2t-1 L η , xj ≤P, yj ≤P M(x,y) =0 Λ(x,y)=0 d M(x, y) l e δΩ(M(x,y)) P 2t-1 L η . (3.3)
Proof. The first estimate in (3.2) is a special case of Theorem 3 of Hua [START_REF] Hua | Additive theory of prime numbers[END_REF], and the second can be established by a development of the underlying method (see also Lemma 1 of Vaughan [START_REF] Vaughan | On Waring's problem for smaller exponents[END_REF]). To verify the first bound in (3.3), substitute y t = t-1 j=1 (x j -y j ) + x t into M (x, y). Then the sum on the left hand side of (3.3) does not exceed the sum x1,...,xt≤P y1,...,yt-1≤P

d t-1 j=1 (x k j -y k j ) + x k t - t-1 j=1 (x j -y j ) + x t k l
, and the desired bound again follows from Theorem 3 of Hua [START_REF] Hua | Additive theory of prime numbers[END_REF]. Finally, once again a development of the method also yields the second estimate in (3.3).

We now consider the pair of diophantine equations

(3.4) x k 1 + x k 2 + x k 3 = y k 1 + y k 2 + y k 3 , x 1 + x 2 + x 3 = y 1 + y 2 + y 3 .
Let Γ(P ) denote the number of solutions of (3.4) with 1 ≤ x j ≤ P , 1 ≤ y j ≤ P such that x 1 , x 2 , x 3 is not a permutation of y 1 , y 2 , y 3 . Then, by Vaughan and Wooley [START_REF] Vaughan | On a certain nonary cubic form and related equations[END_REF] when k = 3, and by Greaves [START_REF] Greaves | Some Diophantine equations with almost all solutions trivial[END_REF] when k ≥ 4, there exists γ > 0 such that Γ(P ) P 3-γ . With this value of γ, this immediately implies the following estimate.

Lemma 4. For U ⊂ {1, 2, . . . , [P ]}, let U (P ) denote the number of solutions of (3.4) with x j ∈ U , y j ∈ U . Then

U (P ) #U 3 + P 3-γ .
The next lemma should be compared with the classical lemma of Hua in the theory of Waring's problem. Lemma 5. There is a positive number η such that whenever 2 ≤ j ≤ k, then

1 0 1 0 |f (α, β)| 2 j +2 dα dβ P 2 j -j+1 L η .
Proof. Let j = 2. By orthogonality, the integral equals U (P ) with U = [1, P ]∩Z so that this case of Lemma 5 is a consequence of Lemma 4.

We now suppose that the conclusion of Lemma 5 has been established for a particular value of j with 2 ≤ j < k, and proceed by induction. Recall that J = 2 j-1 , and return to (2.12). Repeated Weyl differencing via Cauchy's inequality then gives

(3.5) |f (α, β)| 2J P 2J-j-1 |h1|<P . . . |hj |<P x∈I(h) e αh 1 . . . h j Q h (x)
where

I(h) ⊂ [1, P ] is a suitable interval and Q h ∈ Z[X] has degree k -j. Now let r(l) denote the number of solutions of (3.6) h 1 . . . h j Q h (x) = l
with all variables h 1 , . . . , h j , x subject to the summation conditions in the preceding display. Then

|f (α, β)| 2J P 2J-j-1 l∈Z r(l)e(-αl).
By (1.4),

|f (α, β)| 2J+2 = xi≤P, yi≤P 1≤i≤J+1 e αM(x, y) + βΛ(x, y)
where Λ and M are the forms defined in (3.1) with t = J + 1. Hence, by orthogonality, (3.7)

1 0 1 0 |f (α, β)| 4J+2 dα dβ P 2J-j-1
x,y Λ(x,y)=0 r M(x, y) .

One has r(0) P j , and r(l) d(|l|) j+1 for l = 0. Hence the contribution to (3.7) of all terms where M(x, y) = 0 can be estimated by (3.3), and does not exceed P 4J-j L η . This leaves the solutions of M(x, y) = Λ(x, y) = 0, and by orthogonality again, these contribute to (3.7) at most

P 2J-1 1 0 1 0 |f (α, β)| 2J+2 dα dβ P 4J-j L η .
This completes the induction step, and the proof of the lemma.

Let κ denote a positive number sufficiently large in terms of k, and write The next lemma is a considerable refinement of the preceding lemma, and crucial for all later work.

(3.8) D = L κ , D = P 1/(6k) . Let E denote
Lemma 6. Let 2 ≤ j ≤ k. Then 1 0 1 0 |h(α, β)| 2 j +2 dα dβ P 2 j -j+1 L ε-1-1 2 j(j-1) .
Proof. When j = 2, this follows from orthogonality, Lemma 4 and (3.9). Now suppose the estimate is known for a particular value of j with 2 ≤ j < k, and proceed by induction. The argument to follow is very similar to that on pp. 14-19 of [START_REF] Vaughan | On Waring's problem for smaller exponents[END_REF], so we shall be brief whenever the modifications to [START_REF] Vaughan | On Waring's problem for smaller exponents[END_REF] are evident.

We begin by applying Weyl differencing to (3.10). The first differencing is performed as in (2.12), and delivers the initial inequality

|h(α, β)| 2 ≤ #E + 0<|h|<P x∈E x+h∈E e α((x + h) k -x k ) .
Since j ≥ 2, we have to difference further to reach the inequality

|h(α, β)| 2 j P 2 j -j-1 #E j + P 2 j -j-1 1≤|hi|<P 1≤i≤j x e αh 1 . . . h j Q j (x, h)
in which the sum over x is subject to the constraints x ∈ E and x + h i ∈ E (1 ≤ i ≤ j), and Q j is as in (3.5). Now multiply with |h(α, β)| 2 j +2 and integrate. Then, recalling that J = 2 j-1 , one finds that

1 0 1 0 |h(α, β)| 4J+2 dα dβ (3.11) P 2J-j-1 #E j 1 0 1 0 |h(α, β)| 2J+2 dα dβ + P 2J-j-1 x,y Λ(x,y)=0 r M(x, y)
where r(l) has the same meaning as in (3.6), M and Λ are defined by (3.1) with t = J + 1, and x, y are subject to x j , y j ∈ E , M(x, y) = 0. By the induction hypothesis, the first term on the right hand side does not exceed

P 2J-1 L ε-j P 2J-j+1 L ε-1-1 2 j(j-1) P 4J-j L ε-1-1 2 j(j+1) ,
as required. In view of (3.11), the induction will be complete once the inequality (3.12)

x,y

Λ(x,y)=0 r |M(x, y)| P 2J+1 L ε-1-1 2 j(j+1)
is established; here and later the sum is subject to same conditions as in (3.11).

Let D = L κ be as in (3.8). The contribution to (3.12) arising from summands with |M(x, y)| ≤ (P/D) k is small. To see this, we use Cauchy's inequality to infer that (3.13)

Λ(x,y)=0 1≤|M(x,y)|≤(P/D) k r(|M(x, y)|) ≤ T 1 T 2 1/2
where

T 1 = Λ(x,y)=0 d(|M(x, y)|) 2j
and T 2 denotes the number of all 1 ≤ x i , y i ≤ P , 1 ≤ i ≤ J + 1 with

(3.14) Λ(x, y) = 0, 1 ≤ |M(x, y)| ≤ (P/D) k .
By (3.3), we have T 1 P 2J+1 L η . With any solution x, y counted by T 2 , we associate the numbers

u = J+1 i=2 (y i -x i ), w = J+1 i=2 (y k i -x k i ).
Let T 2 denote the number of solutions counted by T 2 where 0 ≤ u ≤ P/D, and let T 2 denote the number of those solutions counted by T 2 where P/D < u ≤ JP . Then, by symmetry in x and y, it follows that

T 2 ≤ 2(T 2 + T 2 ).
To estimate T 2 , we consider one particular choice of x 2 , . . . , x J+1 , y 3 , . . . , y J+1 ; there are O(P 2J-1 ) possibilities for this. The conditions that 0 ≤ u ≤ P/D leaves O(P/D) choices for y 2 . Once these variables are fixed, u is also fixed, and so x 1 -y 1 = u leaves O(P ) choices for the pair x 1 , y 1 . This shows that T 2 P 2J+1 /D. The initial treatment of T 2 is similar. Fix one of the O(P 2J ) choices for x 2 , . . . , x J+1 , y 2 , . . . , y J+1 with u > P/D. This fixes u and w, and by (3.14) it remains to count the x 1 , y 1 with x 1 -y 1 = u, |x k 1 -y k 1 -w| ≤ (P/D) k . We eliminate x 1 , and consider the inequality

(3.15) |(y 1 + u) k -y k 1 -w| ≤ (P/D) k .
Let z 1 and z 2 be two solutions (for y 1 ) of (3.15) with 1 ≤ z 1 ≤ z 2 ≤ P . Then

(3.16) (z 2 + u) k -z k 2 -(z 1 + u) k + z k 1 ≤ 2(P/D) k .
A direct computation yields

(z 2 + u) k -z k 2 -(z 1 + u) k + z k 1 = k(k -1) u 0 z2-z1 0 (z 1 + ζ + ξ) k-2 dζ dξ ≥ k(k -1) u 0 z2-z1 0 ζ k-2 dζ dξ = ku(z 2 -z 1 ) k-1 .
Recalling

Λ(x,y)=0 |M(x,y)|≤(P/D) k r(|M(x, y)|) P 2J+1 L 1 2 (η-κ) .
We are reduced to estimating Let (n) denote the number of solutions of the system |M(x, y)| = n, Λ(x, y) = 0 with x i , y i ∈ E (1 ≤ i ≤ J +1). Then (n) = 0 for n > (J +1)P k , and the definition of r(n) shows that for (P/D) k < n ≤ 2JP k one has r(n)

(log log P ) j ∆ j+1 (n). It follows that (3.19) Υ L ε n ∆ j+1 (n) (n).
Let ν be a (small) positive number, and let δ be as in Lemma 3. Let

Y = exp δνL (η + k 2 ) log L . Let M = {n ∈ N : p | n ⇒ p ≤ Y }, N = {n ∈ N : p | n ⇒ p > Y }. Then, any n ∈ N has a unique factorisation n = n * n † with n * ∈ M , n † ∈ N . Note that n * > P ν implies Ω(n) log Y ≥ log n * > νL so that δΩ(n) > (η + k 2 ) log L. By (3.3), (3.20) n * >P ν ∆ j+1 (n) (n) n d(n) 2j+2 e δΩ(n) (n)L -η-k 2 P 2J+1 L -k 2
which is acceptable. For the complementary portion of (3.19), we have

n * ≤P ν ∆ j+1 (n) (n) ≤ m∈M m≤P ν n∈N ∆ j+1 (mn) (mn) ≤ m∈M m≤P ν ∆ j+1 (m) n∈N d j+1 (n) (mn).
We now require the following simple observation.

Lemma 7. Let k ≥ 3 and ν > 0. There exists a number L 0 depending only on k and ν with the property that for any 2 ≤ j ≤ k -1 and any n < 2JP k there is a divisor n 1 of n with n 1 ≤ P ν and d j+1 (n) j L0Ω(n1) .

Although not highlighted as a lemma, the conclusion of Lemma 7 is established inter alia in [START_REF] Vaughan | On Waring's problem for smaller exponents[END_REF], starting on p. 16 after [27, (3.15)].

We apply Lemma 7 to the inner sum in the previous display and obtain (3.21)

n * ≤P ν ∆ j+1 (n) (n) m∈M m≤P ν ∆ j+1 (m) n1∈N n1≤P ν j L0Ω(n1) n≡0 mod mn1 (n).
It will be convenient to write d = mn 1 . Then, the conditions active in (3.21) imply that d ≤ P 2ν ≤ √ P . Further, the sum n≡0 mod d (n) does not exceed the number of solutions of

(3.22) J+1 i=1 (x i -y i ) = 0, J+1 i=1 (x k i -y k i ) ≡ 0 mod d with x i , y i ∈ E (1 ≤ i ≤ J + 1)
. Let a, b be a solution of the pair of congruences

(3.23) J+1 i=1 (a i -b i ) ≡ J+1 i=1 (a k i -b k i ) ≡ 0 mod d and choose x i , y i ∈ E with x i ≡ a i mod d for 1 ≤ i ≤ J + 1 and y i ≡ b i mod d for 1 ≤ i ≤ J.
Then determine y J+1 through the linear equation in (3.22). By (3.23), it follows that y J+1 ≡ b J+1 mod d, and by (3.9), we infer that the number of solutions to (3.22) with x i , y i ∈ E and

x i ≡ a i mod d, y i ≡ b i mod d (1 ≤ i ≤ J + 1) does not exceed O (P/d) 2J+1 L ε-1-2J . We conclude that n : d|n (n) P 2J+1 L ε-1-2J d -1-2J S(d)
where S(d) is the number of incongruent solutions to the pair of congruences (3.23). Since S(d) is multiplicative, we deduce from (3.21) that (3.24)

n * ≤P ν ∆ j+1 (n) (n) P 2J+1 L ε-1-2J Ξ 1 Ξ 2 in which Ξ 1 = m∈M m≤P ν ∆ j+1 (m)S(m) m 1+2J , Ξ 2 = n∈N n≤P ν j L0Ω(n) S(n) n 1+2J .
Further progress depends on upper bounds for S(d) that we now derive. By (2.1) and orthogonality,

(3.25) S(d) = d -2 d a=1 d b=1 |S(q, a, b)| 2J+2 .
But j ≥ 2, so that 2J + 2 ≥ 6, and then

S(d) ≤ d 2J-6 d a=1 d b=1 |S(q, a, b)| 6 = d 2J-4 S 0 (d)
where S 0 (d) is the number of solutions of the congruences

3 i=1 (u i -v i ) ≡ 3 i=1 (u k i -v k i ) ≡ 0 mod d.
In particular, S 0 (d) is a multiplicative function. We now have

Ξ 1 ≤ m∈M m≤P ν ∆ j+1 (m)S 0 (m) m 5 , Ξ 2 ≤ n∈N n≤P ν j L0Ω(n) S 0 (n) n 5 .
By (2.1), whenever d = (q; a; b), one has S(q, a, b) = dS(q/d, a/d, b/d). Hence, on noting that S 0 (d) equals the right hand side of (3.25) with J = 2, and then collecting terms according to (q; a; b), one readily confirms that

(3.26) S 0 (m) m 4 = q|m A(q)
where (3.27) A(q) = q -6 q a,b=1 (a;b;q)=1 |S(q, a, b)| 6 .

By (3.26) and Möbius inversion, A(q) is multiplicative. We shall prove momentarily that the series

(3.28) ∞ q=1 A(q)d j+1 (q)
q , q∈N j L0Ω(q) A(q) q converge, and that the second series (which depends on P ) is bounded above by a constant depending only on k. Once this is established, the proof of Lemma 6 is swiftly completed. Indeed, the familiar inequality ∆ j+1 (uw

) ≤ ∆ j+1 (u)d j+1 (w) now implies that Ξ 1 ≤ m∈M m≤P ν ∆ j+1 (m) m q|m A(q) ≤ u≤P ν ∆ j+1 (u) u q≤P ν A(q)d j+1 (q) q ,
and by (3.28) the sum over q is bounded. Also, by Hall and Tenenbaum [START_REF] Hall | Divisors. Cambridge Tracts in Mathematics[END_REF], Theorem 70, the sum over

u is O(L 1+ε ), whence Ξ 1 L 1+ε . Similarly, Ξ 2 ≤ n∈N n≤P ν j L0Ω(n) n q|n A(q) ≤ u∈N u≤P ν j L0Ω(u) u q∈N q≤P ν j L0Ω(q) A(q) q .
Here again, the sum over q is bounded and

u∈N u≤P ν j L0Ω(u) u = Y <p≤P ν 1 + ∞ l=1 j L0l p l (log log P ) j L 0 , so that Ξ 2 L ε .
On collecting together, it follows that the expression on the left hand side of (3.24) is O P 2J+1 L ε-2J . But 2J ≥ 1 + 1 2 j(j + 1) for j ≥ 2, and therefore, by (3.24), (3.19) and (3.20), the sums (3.18) and (3.17) (with κ = η + k 2 ) are both sufficiently small to imply (3.12). This completes the proof of Lemma 6.

In preparation for the discussion of the series in (3.28), we require an upper bound for A(q). By multiplicativity, it will suffice to consider the case where q = p l is a prime power. In this case, one infers from (3.27) and Lemma 2 that

(3.29) A(p l ) = p -6l p l a=1 p a p l b=1 |S(p l , a, b)| 6 .
For a crude bound, note that whenever p a one has p -l S(p l , a, b) p -l/k . This is slighly stronger than (2.1) but follows from the proof of [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF], Theorem 7.1, or Chalk [START_REF] Chalk | On Hua's estimates for exponential sums[END_REF], for example. By (3.29), it follows that

A(p l ) p -2l-4l/k p l a=1 p l b=1 |S(p l , a, b)| 2 ,
and by orthogonality, one derives the estimate

(3.30) p -l A(p l ) p -4l/k .
We proceed to establish the alternative estimate

(3.31) p -l A(p l ) p -2
that is valid for all primes p with p k. Indeed, by Lemma 4.3 of Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF], one has S(p, a, 0) p 1/2 whenever p a, and when p b, Lemma 4.1 of Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF] gives S(p, a, b) p 1/2 . By (3.29), it follows that A(p) p -1 . This already confirms (3.31) when l = 1, and by (3.26), one also finds that

(3.32) S 0 (p) p 4 .
Now let l ≥ 2, and consider a solution of the system of congruences

(3.33) 3 i=1 (u i -v i ) ≡ 3 i=1 (u k i -v k i ) ≡ 0 mod p l with 1 ≤ u i , v i ≤ p l .
Such a solution is said to be non-singular modulo p if the array ku k-1

1 ku k-1 2 ku k-1 3 -kv k-1 1 -kv k-1 2 -kv k-1 3 1 1 1 -1 -1
-1 has rank 2, modulo p, and otherwise singular modulo p. Note that for p k, a solution is singular modulo p if and only if

(3.34) u k-1 1 ≡ u k-1 2 ≡ u k-1 3 ≡ v k-1 1 ≡ v k-1 2 ≡ v k-1 3 mod p.
It follows that there are at most (k -1) 4 p 5l-4 singular solutions of (3.33), because for each of the p l choices for u 1 , the remaining variables will satisfy (3.34), leaving at most (k -1) 4 p 4l-4 possibilities for u 2 , u 3 , v 1 , v 2 by the theory of power residues. The value of v 3 mod p l is then fixed through (3.33).

When p k, we count the non-singular solutions to (3.33) with u k-1

1 ≡ v k-1
1 mod p with the aid of Hensel's Lemma. Indeed, each such solution, when reduced modulo p, corresponds to one of the O(p 4 ) solutions counted by S 0 (p) , and at most O(p 4l-4 ) of them will reduce to the same solution modulo p, because for a solution u, v of the system

u k -v k ≡ a mod p, u -v ≡ b mod p with p u k-1 -v k-1 , there is exactly one pair u 1 , v 1 with 1 ≤ u 1 , v 1 ≤ p l and u k 1 -v k 1 ≡ a mod p l , u 1 -v 1 ≡ b mod p l .
A similar argument applies for counting non-singular solutions with u k-1 1 ≡ u k-1 2 mod p, so that, by symmetry, there are at most O(p 4l ) non-singular solutions of (3.33). It follows that S 0 (p l ) p 4l + p 5l-4 . We now ignore the condition p a in (3.29) and use orthogonality to deduce that

A(p l ) p -4l S 0 (p l ) 1 + p l-4 .
For l ≥ 2, this contains (3.31), as required.

We are ready to discuss the first of the two series in (3.28). The easy bound d j+1 (p l ) ≤ (l + 1) j coupled with (3.30) suffices to recognise the sum

(3.35) ∞ l=0 d j+1 (p l ) A(p l ) p l
as a convergent one, and if one uses (3.31) for p > k ≥ l ≥ 1, and (3.30) for l > k, then this sum is seen to be of the form 1 + O(p -2 ). The sums (3.35) are the factors in the Euler product for the sum (3.28), so that the latter indeed converges.

A similar argument applies to the second sum in (3.28). Rewritten as an Euler product, this sum becomes

(3.36) p>Y ∞ l=0 j L0l A(p l ) p l .
When P is sufficiently large, one has j L0 p -4/k < 1/2 for all p > Y , so that (3.30) yields

l>k j L0l A(p l ) p l l>k j L0 p 4/k l p -4 .
On using (3.31) for 1 ≤ l ≤ k as in the previous discussion, one again finds that the Euler factors in (3.36) are of the type 1 + O(p -2 ), thus confirming the claims concerning the second sum in (3.28).

The principal proposition

Our next result is a version of Hua's lemma with a logarithmic saving, similar to Theorem B of Vaughan [START_REF] Vaughan | On Waring's problem for smaller exponents[END_REF].

Lemma 8. Let k ≥ 3, K = 2 k-1 and t = 3 2 K + 2. Then 1 0 1 0 |f (α, β)| t dα dβ P t-k-1/2 L ε-3 .
Note that Theorem 2 follows on combining the conclusions of Lemmas 1 and 8. The proof of Lemma 8 will occupy this and the next two sections. Only the initial steps of the proof work for all values of k. Let I denote the integral that is estimated in Lemma 8. On noting that t is the arithmetic mean of K + 2 and 2K + 2, one deduces from Lemma 5 and Schwarz's inequality that (4.1)

I P t-k-1/2 L η .
A similar argument may be applied to the exponential sum h(α, β). An application of Schwarz's inequality combined with Lemma 6 yields (4.2)

1 0 1 0 |h(α, β)| t dα dβ P t-k-1/2 L ε-1-1 2 (k-1) 2
.

We now apply a differencing argument that reduces the estimation of I to that of the integral considered in (4.2). The main ideas are adopted from Vaughan [START_REF] Vaughan | On Waring's problem for smaller exponents[END_REF]. By orthogonality, I is the number of solutions of the pair of equations

(4.3) x k -y k = 3K/2 i=1 (x k i -y k i ), x -y = 3K/2 i=1 (x i -y i ) with x, y, x i , y i all constrained to the interval [1, P ]. Also, when U is a subset of {(x, y) ∈ N 2 : 1 ≤ x,
I ≤ 4 1 0 1 0 |h(α, β)| 2 |f (α, β)| t-2 dα dβ ≤ 4 1 0 1 0 |h(α, β)| t dα dβ 2/t 1 0 1 0 |f (α, β)| t dα dβ 1-2/t
.

Here Hölder's inequality was used to infer the second inequality. The second integral on the right is I, and therefore

I ≤ 2 t 1 0 1 0 |h(α, β)| t dα dβ.
Hence in this case, the desired estimate for I is a consequence of (4.2). Next, suppose that I ≤ 4I(A ). We define and sort the pairs (x, y) ∈ A according to the value of d = (x; y). Then, by orthogonality, a consideration of the underlying diophantine equations reveals that

I(A ) ≤ d>D 1 0 1 0 |f (αd k , βd; P/d)| 2 |f (α, β)| t-2 dα dβ.
The initial assumption and Hölder's inequality yield

I ≤ 4 d>D 1 0 1 0 |f (αd k , βd; P/d)| t dα dβ 2/t I 1-2/t .
We use the bound for I provided by (4.1), and for the remaining integral on the right hand side here, a consideration of the underlying diophantine equations shows that (4.1) again supplies a bound, this time with P/d in place of P . One then finds that

I P t-k-1/2 L η d>D d -(2t-2k-1)/t P t-k-1/2 L η D -1/8 .
Hence for κ ≥ 8(η + k 2 ), this shows that I P t-k-1/2 L -k 2 which is acceptable. It remains to consider the case where I ≤ 4I(B). The initial steps are along familiar lines. Recall the definition of B and sort the solutions of (4.3) counted by I(B) according to the value of p = m(x). Then D < p ≤ D , and the condition that (x; y) ≤ D implies p y. Hence I(B) does not exceed the number of solutions to the equation

(4.6) (pw) k -y k = 3K/2 i=1 (x k i -y k i ), pw -y = 3K/2 i=1 (x i -y i )
in primes p with D < p ≤ D and natural numbers w, y, x i , y i satisfying w ≤ P/p and (4.7) y ≤ P, x i ≤ P, y i ≤ P, p y.

Let I M denote the number of solutions of (4.6) constrained to (4.7) and M < p ≤ 2M , w ≤ P/M . Then, on splitting the range for p into dyadic intervals, one finds that there is some M with D ≤ M ≤ D and (4.8)

I ≤ 4I(B) LI M .
We now recall (4.5) and write

f p (α, β) =
g(p k α,pβ)f p (-α, -β)|f (α, β)| t-2 dα dβ ≤ Θ 1/(K+2) p I 1-(2/t) 1 0 1 0 |f p (α, β)| t dα dβ K+4 t(2K+4) .
On considering the underlying diophantine equations, it is immediate that the integral on the far right is bounded by I. Hence

1 0 1 0 g(p k α, pβ)f p (-α, -β)|f (α, β)| t-2 dα dβ Θ 1/(K+2) p I (K+1)/(K+2) .
By orthogonality, the integral on the left is non-negative. We may sum over p to first infer from (4.9) that

I M (IM ) (K+1)/(K+2) M <p≤2M Θ p 1/(K+2)
, and then, by (

I L K+2 M K+1 M <p≤2M Θ p . 4.8), (4.11)
The next step is to show that whenever D ≤ M ≤ D then (4.12)

M <p≤2M Θ p P t-k-1/2 M -K-3/2 L η .
Once this is established, it suffices to recall that D = L κ , and to combine (4.11) and (4.12) to finally conclude that I P t-k-1/2 L -k 2 holds in this last case as well, provided only that κ is large enough.

We shall estimate the sum in (4.12) by a differencing argument. When k ≥ 4, the problem at hand can be approached by combining ideas contained in Vaughan [START_REF] Vaughan | On Waring's problem for smaller exponents[END_REF] and Wooley [START_REF] Wooley | On simultaneous additive equations[END_REF]. The rather technical details are provided in the next section. For k = 3 this argument collapses, and we present an alternative approach via the Hardy-Littlewood method in the following section.

Efficient differencing

Throughout this section we suppose that k ≥ 4. Then K/4 ≥ 2, and K/4 is even. We will use this frequently. We prepare for the differencing operation with a technical estimate concerning certain congruences. For p > k, let Z p (a, b) be the set of solutions z = (z 1 , z 2 , . . . , z K/4 ) to the simultaneous congruences (5.1)

z k 1 + z k 2 + . . . + z k K/4 ≡ a mod p k , z 1 + z 2 + . . . + z K/4 ≡ b mod p with 1 ≤ z i ≤ p k and p z i for all 1 ≤ i ≤ K/4. Also, let Z p = max a,b #Z p (a, b). Lemma 9. Let p > k ≥ 4. Then Z p p 1 4 Kk-k-1 .
Proof. First suppose that k = 4. Then K/4 = 2. Consider the solutions w 1 , w 2 of (5.2)

w 4 1 + w 4 2 ≡ a mod p, w 1 + w 2 ≡ b mod p with 1 ≤ w i ≤ p -1 (i = 1, 2
). Here, one may eliminate w 2 . Then w 1 satisfies a polynomial congruence of degree 4 which has at most 4 solutions. For any solution z of z 4 1 + z 4 2 ≡ a mod p 4 , z 1 + z 2 ≡ b mod p that is counted by Z p , there is a solution w of (5.2) with z i ≡ w i mod p. However, for a fixed solution w of (5.2), there are p 3 choices for z 2 with 1 ≤ z ≤ p 4 and z 2 ≡ w 2 mod p, and for any such z 2 , there are at most four solutions of z 4 1 + z 4 2 ≡ a mod p 4 with p z 1 and 1 ≤ z 1 ≤ p 4 . This shows that Z p ≤ 16p 3 , as required.

Next, suppose that k ≥ 5. Let Z p (a , b ) denote the set of solutions z 1 , z 2 , z 3 of the congruences

(5.3) z k 1 + z k 2 + z k 3 ≡ a mod p k , z 1 + z 2 + z 3 ≡ b mod p with p z 1 z 2 z 3 and 1 ≤ z i ≤ p k for i = 1, 2, 3. The bound (5.4) #Z p (a , b ) p 2k-1
holds uniformly for a, b ∈ Z, p > k. This can be seen as follows. In the system of congruences (5.5)

w k 1 + w k 2 + w k 3 ≡ a mod p, w 1 + w 2 + w 3 ≡ b mod p one may eliminate w 3 . But w k 1 + w k 2 + (b -w 1 -w 2 )
k is a polynomial in w 1 , and when 2 | k, the degree is k and the leading coefficient is 2. Hence we find at most kp incongruent solutions of (5.5). When k is odd, and w 2 ≡ b mod p, then the degree is k -1 and the leading coefficient is k(b -w 2 ), so that (5.5) can have at most (k -1)(p -1) solutions with w 2 ≡ b mod p, and further p solutions with w 2 ≡ b mod p. Hence in all cases, there are at most kp solutions. Any solution of (5.3) reduces to one of (5.5). There are p 2k-2 choices of z 2 , z 3 mod p k with z i ≡ w i mod p. Now solve for z 1 from the first congruence in (5.3). Since p z 1 , there are at most k solutions for z 1 . This confirms (5.4). To complete the proof of the lemma, it now suffices to take (5.4), and to sum over z 4 , . . . , z K/4 trivially.

a = a -z k 4 -. . . -z k K/4 , b = b -z 4 -. . . -z K/4 in
We now return to the main theme. Let p be a prime with M < p ≤ 2M . The goal is to estimate Θ p , as defined in (4.10). By orthogonality, Θ p is the number of solutions to the pair of equations (5.6)

K/4 i=1 (x k i -y k i ) = p k 1+K/2 i=1 (u k i -v k i ), K/4 i=1 (x i -y i ) = p 1+K/2 i=1 (u i -v i )
in natural numbers x i , y i , u i , v i constrained to (5.7)

x i ≤ P, y i ≤ P, u i ≤ P/M, v i ≤ P/M, p x i y i (1 ≤ i ≤ K/4).
Note that solutions to (5.6) satisfy

(5.8)

x k 1 + x k 2 + . . . + x k K/4 ≡ y k 1 + y k 2 + . . . + y k K/4 mod p k , x 1 + x 2 + . . . + x K/4 ≡ y 1 + y 2 + . . . + y K/2 mod p.
The following argument is an elementary variant of a very similar exponential sum technique underpinning the proof of Lemma 2.2 in Wooley [START_REF] Wooley | On simultaneous additive equations[END_REF]. For given data n, m, z 1 , . . . , z K/4 , let Φ p (z, n, m) denote the number of solutions to

K/4 i=1 x k i + p k 1+K/2 i=1 u k i = n, K/4 x i + p 1+K/2 i=1 u i = m
satisfying the relevant conditions in (5.7) and

x i ≡ z i mod p k , for 1 ≤ i ≤ K/4.
Then, by (5.8) and the discussion preceding this observation,

Θ p = ∞ n,m=-∞ p k a=1 p b=1 z∈Zp(a,b) Φ p (z, n, m) 2 .
By Cauchy's inequality and Lemma 9,

Θ p ≤ ∞ n,m=-∞ p k a=1 p b=1 Z p z∈Zp(a,b) Φ p (z, n, m) 2 p 1 4 Kk-k-1 ∞ n,m=-∞ z mod p k Φ p (z, n, m) 2 p 1 4 Kk-k-1 Ψ p (5.9)
where Ψ p is the number of solutions to (5.6) and (5.7) with the additional constraints that (5.10)

x i ≡ y i mod p k (1 ≤ i ≤ K/4). Now let (5.11) G(α, β) = x≤P,y≤P x≡y mod p k e α(x k -y k ) + β(x -y) .
By orthogonality, (5.12)

Ψ p ≤ 1 0 1 0 G(α, β) K/4 |g(αp k , βp)| K+2 dα dβ.
By (5.11),

G(α, β) = [P ] + 2Re y<x≤P x≡y mod p k e α(x k -y k ) + β(x -y) which implies that (5.13) |G(α, β)| K/4 P K/4 + y<x≤P x≡y mod p k e α(x k -y k ) + β(x -y) K/4 .
We use this in (5.12) and apply Lemma 5 to deduce that (5.14) Ψ p Ψ p + P K/4 (P/M ) K+2-k L η

where

Ψ p = 1 0 1 0 y<x≤P x≡y mod p k e α(x k -y k ) + β(x -y) K/4 |g(p k α, pβ)| K+2 dα dβ.
By orthogonality, Ψ p is the number of solutions of the pair of equations

K/4 i=1 (-1) i (x k i -y k i ) = p k 1+K/2 i=1 (u k i -v k i ), K/4 i=1 (-1) i (x i -y i ) = p 1+K/2 i=1 (u i -v i
subject to (5.7), (5.10) and x i > y i . We write

z i = x i + y i , h i = p -k (x i -y i ).
Then, since 2x i = z i + p k h i and 2y i = z i -p k h i , one finds that

2 k (x k i -y k i ) = p k ϕ p (z i , h i ) where ϕ p is the integral polynomial ϕ p (z, h) = p -k (z + hp k ) k -(z -hp k ) k ,
and we may then conclude that Ψ p does not exceed the number of solutions of the system

K/4 i=1 (-1) i ϕ p (z i , h i ) = 2 k 1+K/2 i=1 (u k i -v k i ), p k-1 K/4 i=1 (-1) i h i = 1+K/2 i=1 (u i -v i )
in which the variables are subject to

1 ≤ h i ≤ P M -k , z i ≤ 2P, u i ≤ P/M, v i ≤ P/M.
Now write H = P M -k and introduce the exponential sum

F p (α, β) = h≤H z≤2P
e αϕ p (z, h)

+ βhp k-1 .
Then, once again by orthogonality, (5.15)

Ψ p ≤ 1 0 1 0 |F p (α, β)| K/4 |g(2 k α, β)| K+2 dα dβ.
We have now completed the first differencing step. The differencing was efficient because the congruences (5.8) reduce the potential reservoir for the variables x i , y i by a factor p -k-1 , and one recovers this through Lemma 9. We proceed by taking further differences. By Cauchy's inequality,

|F p (α, β)| 2 ≤ H h≤H z≤2P e αϕ p (z, h) 2 .
Note that β is absent from the right hand side. Thus, we may proceed as with the usual proof of Weyl's inequality (see Lemma 2.4 of Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF]) to confirm that there are certain natural numbers c(l) (depending also on p which we suppress) with c(0

) P k-1 M -k , c(l) d k (l) (l = 0
), and such that (5.16)

|F p (α, β)| K/2 ≤ P K-k M (1-1 2 K)k l c(l)e(αl).
At this point it might be worth recalling that in the current context one has k ≥ 4, and that therefore β is indeed absent from this bound, for all k under consideration.

To bound the integral

V = 1 0 1 0 |F p (α, β)| K/2 |g(2 k α, β)| K+2 dα dβ,
one inserts the inequality (5.16), and then separates off the arising from l = 0. This term will contribute to V at most

P K-k M (1-1 2 K)k c(0) 1 0 1 0 |g(2 k α, β)| K+2 dα dβ.
We may apply Lemma 5 to the integral on the right, and then conclude that this contribution to V does not exceed

(P K-k M (1-1 2 K)k )(P k-1 M -k )(P/M ) K+2-k L η P 2K-k+1 M (1-1 2 K)k-K-2 L η .
By orthogonality, the terms that correspond to l = 0 produce a term not exceeding

P K-k M (1-1 2 K)k ui,vi d k u k 1 -v k 1 + . . . + u k 1 2 K+1 -v k 1 2 K+1
in which the variables are restricted by u i ≤ P/M , v i ≤ P/M and

u 1 -v 1 + . . . + u 1 2 K+1 -v 1 2 K+1 = 0. By Lemma 3, this does not exceed (P K-k M (1-1 2 K)k )(P/M ) K+1 L η P 2K-k+1 M (1-1 2 K)k-1-K L η . Collecting together yields V P 2K-k+1 M (1-1 2 K)k-1-K L η
. By (5.15) and Hölder's inequality,

Ψ p ≤ V 1/2 1 0 1 0 |g(2 k α, β)| K+2 dα dβ 1/2
, and another use of Lemma 5 produces (5.17)

Ψ p P 3 2 K-k+ 3 2 M 1 2 (1-1 2 K)k-K-3 2 + 1 2 k L η .
By (5.9) and (5.14), we may now conclude that

M <p≤2M Θ p M 1 4 Kk-k (max p Ψ p + P K/4 (P/M ) K+2-k L η ).
As is readily checked, this establishes (4.12), as was required to complete the proof of Lemma 8.

Inefficient differencing

In this section we establish the case k = 3 of Lemma 8. Our approach needs substantial revision because Lemma 4 provides optimal control on the sixth moment of Weyl sums, and if one differences two blocks of two variables beyond this as would be needed for efficient differencing, then one works with ten variables. But when k = 3 one has t = 8. One could, at least in principle, study a tenth moment, but savings can then be expected only if differencing is performed over minor arcs only. This would entail considerable complication in detail, and we prefer an eighth moment for consistency with our work in the previous section. Fortunately, the inevitable loss of a factor M in an inefficient differencing can be restored in part by averaging over the auxiliary prime p. The technique elaborates on ideas of Vaughan [START_REF] Vaughan | On Waring's problem for cubes[END_REF].

We now return to (4.10), temporarily fix a prime p with M < p ≤ 2M , and observe by orthogonality that Θ p equals the number of solutions of the diophantine system (6.1)

x 3 1 + p 3 (y 3 1 + y 3 2 + y 3 3 ) = x 3 2 + p 3 (y 3 4 + y 3 5 + y 3 6 ), x 1 + p(y 1 + y 2 + y 3 ) = x 2 + p(y 4 + y 5 + y 6 )
with variables constrained to

x i ≤ P, y i ≤ P/M, p x 1 x 2 .
Any solution of (6.1) satisfies x 3 1 ≡ x 3 2 mod p 3 and x 1 ≡ x 2 mod p. Since p x 1 x 2 , this implies x 1 ≡ x 2 mod p 3 . By Lemma 4, the number of solutions with x 1 = x 2 amount to at most O(P (P/M ) 3 ). By symmetry, it now suffices to count solutions where x 1 > x 2 . According to the preceding comment, we put x 1 = x 2 + hp 3 with h > 0, and z = x 1 + x 2 . Then (6.1) transforms to

(6.2) h(3z 2 + h 2 p 6 ) = 4(y 3 1 + y 3 2 + y 3 3 -y 3 4 -y 3 5 -y 3 6 ), hp 2 = y 1 + y 2 + y 3 -y 4 -y 5 -y 6 .
Let Ξ denote the number of solutions of (6.2) subject to

z ≤ 2P, y i ≤ P/M, h ≤ H, M < p ≤ 2M
where H = P M -3 . Then, on summing over p, the above argument yields (6.3)

M <p≤2M Θ p ≤ 2Ξ + O(P 4 M -2 ). Let F (α) = z≤2P e(3αz 2 ), E(α, β) = M <p≤2M e(αp 6 + βp 2 )
and recall that g(α, β) = f (α, β, P/M ). Then, by orthogonality, (

Ξ = 1 0 1 0 h≤H F (αh)E(αh 3 , βh)|g(4α, β)| 6 dα dβ. 6.4)
Note the similarity with (5.15). We apply the Hardy-Littlewood method to estimate Ξ. Let δ > 0 be a small parameter to be determined later. Let N be the set of all α ∈ [0, 1] where there are coprime a, q with 0 ≤ a ≤ q, 1 ≤ q ≤ P 1+δ and |qα -a| ≤ P δ-1 H -1 . Let n be the complement of N in [0, 1]. Then, by Lemma 4 of Vaughan [START_REF] Vaughan | On Waring's problem for cubes[END_REF], one has (6.5) sup

α∈n h≤H |F (αh)| 2 HP L 3/2 .
Unlike in [START_REF] Vaughan | On Waring's problem for cubes[END_REF], the actual exponent of L is of no importance for us. Next recall (4.5) to observe that h≤H

|E(αh 3 , βh)| 2 = M <p1,p2≤2M f α(p 6 1 -p 6 2 ), β(p 1 -p 2 ), H .
The contribution of summands with p 1 = p 2 here is O(HM ). We claim that whenever α ∈ n and p 1 = p 2 , then f α(p

-p 6 2 ), β(p 1 -p 2 ), H | > H 3/4 (log H) 2 .
Then, by Lemma 1 (with θ = 7/9), there are coprime numbers b, r with r ≤ H 7/9 and |αr(p 6 1 -p 6 2 ) -b| ≤ H -20/9 , and consequently there is a q with q | r(p 6 1 -p 6 2 ) and an a ∈ Z with |αq -a| ≤ H -20/9 . Hence q ≤ H 7/9 (2M ) 6 = 2 6 P 7/9 M 11/3 , and therefore α ∈ N, as one readily confirms, and as was desired. This establishes (6.6).

By (6.5), (6.6) and Cauchy's inequality, sup α∈n h≤H

F (αh)E(αh 3 , βh) H(P M ) 1/2 L.
We use this in conjunction with Lemma 4 to infer that (6.7)

n 1 0 h≤H F (αh)E(αh 3 , βh)|g(4α, β)| 6 dβ dα H(P M ) 1/2 (P/M ) 3 L.
We now consider the major arcs N. When a, q are coprime with |qα -a| ≤ 1/q, then Lemma 3.1 of Vaughan [START_REF] Vaughan | A new iterative method in Waring's problem[END_REF] asserts that h≤H |F (αh)| 2 P ε P 2 H q + P 2 H|qα -a| + P H + q + P 2 H|qα -a| .

For α ∈ N, there are such a, q with q ≤ P 1+δ and |qα -a| ≤ H -1 P δ-1 , and a short calculation then confirms that (

h≤H |F (αh)| 2 P 2+3δ H q + P 2 H|qα -a| • Moreover, by orthogonality, (6.9)

1 0 |g(4α, β)| 4 dβ = l ψ l e(αl)
in which ψ l equals the number of solutions of the pair of diophantine equations

4(y 3 1 + y 3 2 -y 3 3 -y 3 4 ) = l, y 1 + y 2 -y 3 -y 4 = 0 with y i ≤ P/M (1 ≤ i ≤ 4
). The integral representation shows that the Fourier series in (6.9) takes non-negative values only. Hence we may apply Lemma 2 of Brüdern [START_REF] Brüdern | A problem in additive number theory[END_REF] to conclude that

N h≤H |F (αh)| 2 1 0 |g(4α, β)| 4 dβ dα P 3δ+ε P 1+δ ψ 0 + l ψ l ,
and the bounds ψ 0 (P/M ) 2 and l ψ l (P/M ) 3 are immediate. This gives (6.10)

N h≤H |F (αh)| 2 1 0 |g(4α, β)| 4 dβ dα P 3+5δ M -2 .
Also, on using a trivial bound for E(α, β) and Lemma 5 only, we have When M ≤ P 1/15 , this bound is superior to the one in (6.7). Hence, by (6.7) and (6.4) we conclude that Ξ P 9/2 M -11/2 L. The case k = 3 of Lemma 8 now follows via (6.3).

Pruning

In order to facilitate the major arc analysis within the proof of Theorem 1, the minor arc estimate provided by Theorem 2 is to be augmented by a device that restricts the integration for the linear equation to suitable major arcs as well. Some notation is required to make this precise.

Let 1 ≤ q ≤ P , and let M q denote the union of the intervals {α ∈ [0, 1] : |qα -a| ≤ (4k) -1 P 1-k } with 0 ≤ a ≤ q and (a; q) = 1. Note that these intervals are disjoint, as are the various M q . Let K q = M q × [0, 1], and let K be the disjoint union of the K q with 1 ≤ q ≤ P . Further, when 1 ≤ q ≤ P 1/9 , let L q be the union of the rectangles (α, β) ∈ [0, 1] 2 : α -a q ≤ P 1/9-k , β -b q ≤ P -8/9 with 0 ≤ a ≤ q, 0 ≤ b ≤ q and (a; q) = 1. Then, for 1 ≤ q ≤ P 1/9 , one has L q ⊂ K q . Let L be the union of L q with 1 ≤ q ≤ P 1/9 so that L ⊂ K.

Lemma 10. Let s > 3k. Then K\L |f (α, β)| s dα dβ P s-k-1-1/(9k)+ε .
Proof. Let (α, β) ∈ K. Then there are a unique q ∈ [1, P ] with (α, β) ∈ K q , and unique a, b with 0 ≤ a ≤ q, 0 ≤ b ≤ q with (a; q) = 1 and

|qα -a| ≤ (4k) -1 P 1-k , -1 2 < qβ -b ≤ 1 2 . Define f * : K → C by f * (α, β) = q -1 S(q, a, b)v α -a q , β -b q . Whenever (α, β) ∈ K, one infers from Theorem 3 that f (α, β) = f * (α, β) + O P (k-1)/k+ε . Consequently |f (α, β)| s |f * (α, β)| s + P s-s/k+ε .
The linear measure of M q is O(P 1-k ), whence the planar measure of K is O(P 2-k ). Since s ≥ 3k + 1, it follows that (7.1)

K\L |f (α, β)| s dα dβ K\L |f * (α, β)| s dα dβ + P s-k-1-1/k+ε .
Now consider the integral on the right hand side. First, we estimate the contribution from K q with P 1/9 < q ≤ P which amounts to (7.2)

P 1/9 <q≤P q a=1 (a;q)=1 q b=1 q -s |S(q, a, b)| s ∞ -∞ ∞ -∞ |v(ξ, ζ)| s dξ dζ.
By (2.2), we have the alternative estimates

(7.3) v(ξ, ζ) P (1 + P k |ξ|) -1/k , v(ξ, ζ) P (1 + P |ζ|) -1/k , and consequently ∞ -∞ ∞ -∞ |v(ξ, ζ)| s dξ dζ P s-k-1 .
By (2.2) again, the expression in (7.2) is bounded by

P 1/9 <q≤P q 2-s/k+ε P s-k-1 P s-k-1-(1/9k)+ε
which is acceptable. It remains to consider the contribution from the sets K q \ L q with q ≤ P 1/9 to the integral on the right hand side of (7.1), which in fact does not exceed (7.4)

q≤P 1/9 q a=1 (a;q)=1 q b=1 q -s |S(q, a, b)| s (B 1 + B 2 )
where

B 1 = ∞ -∞ |ξ|>P 1/9-k |v(ξ, ζ)| s dξ dζ, B 2 = ∞ -∞ |ζ|>P -8/9
|v(ξ, ζ)| s dζ dξ. By (7.3), and since s ≥ 3k + 1, one has

B 1 P s ∞ -∞ (1 + P |ζ|) -1-1/k dζ |ξ|>P 1/9-k (2 + P k |ξ|) -2 dξ P s-k-10/9 . (7.5)
A similar reasoning yields the same bound for B 2 . By (2.2), it is now readily seen that the expression in (7.4) does not exceed O(P s-k-10/9 ), and the lemma follows from (7.1).

The proof of Theorem 1

This final section is devoted to the proof of Theorem 1 which we launch by disposing of some simple cases. Throughout this section, let s ≥ 2 k + 2, let a i = 0 for 1 ≤ i ≤ s, and suppose that exactly r of the s numbers b i are non-zero. Since the equations (1.1) have a non-singular real solution, one concludes that r ≥ 1. On renumbering the variables, we may arrange that b i = 0 for 1 ≤ i ≤ r. The system (1.1) now takes the shape (8.1)

a 1 x k 1 + • • • + a r x k r + • • • + a s x k s = 0, b 1 x 1 + • • • + b r x r = 0.
Here, we may clear common factors in the second equation, so that we may further suppose that b is a primitive vector. If r = 1 then x 1 = 0, and we are left with a single equation in 2 k + 1 variables that may be treated by the methods of chapter 2 of Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF].

If , then this single equation in s -1 ≥ 2 k + 1 variables may again be treated by the methods of chapter 2 of Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF], the unconventional size constrained on y being readily absorbed by the classical method that need not be commented on any further here. If

a 1 b k 2 = a 2 b k 1 , one has N (P ) = 2P/ max(|b 1 |, |b 2 |) + O(1) N 0
where N 0 is the number of solutions to the equation a

3 x k 3 + • • • + a s x k s = 0 with |x i | ≤ B. Now s -2 ≥ 2 k . If s -2 > 2 k
, then we may proceed as before to establish an asymptotic formula for N 0 by the methods of chapter 2 of Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF], and this will complete the proof of Theorem 1 in this case. This leaves the case where s -2 = 2 k . In this case, the methods of Vaughan [START_REF] Vaughan | On Waring's problem for smaller exponents[END_REF] (and in particular (1.5)) may be combined with the singular series work in chapter 2 and 4 of Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF] to establish that

N 0 = χ ∞ p χ p P s-k-2 1 + O (log P ) -1
where the Euler product is absolutely convergent. Also, χ p is non-zero if and only if a 3 x k 3 + • • • + a s x k s = 0 has a no trivial solution in Q p , and χ ∞ is non-zero if and only if a 3 , a 4 , . . . , a s are not all of the same sign. On collecting together, this establishes Theorem 1 in the case r = 2.

This leaves the case r ≥ 3. Here we define (8.2)

F (α, β) = s i=1 f (a i α, b i β),
and observe that the integral

(8.3) N + a,b (P ) = 1 0 1 0 F (α, β) dα dβ
counts the solutions of (1.1) with 1 ≤ x i ≤ P (i = 1, . . . , s).

In preparation for an application of the Hardy-Littlewood method to the integral (8.3), we define the major arcs W as the union of the boxes (8.4) {(α, β) ∈ [0, 1] 2 : α -a q ≤ P (1/8)-k , β -b q ≤ P -7/8 } with 0 ≤ a ≤ q, 0 ≤ b ≤ q, (a; b; q) = 1 and q ≤ P 1/8 . This union is disjoint. When (α, β) ∈ W is in the box (8.4), we put

f * i (α, β) = q -1 S(q, aa i , bb i )v a i (α -a/q), b i (β -b/q) .
Note that Theorem 3 is not a suitable tool to compare f (a i α, b i β) with f * i (α, β) because the condition (a; q) = 1 is not met on some boxes. However, Theorem 7.2 of Vaughan [START_REF] Vaughan | The Hardy-Littlewood Method[END_REF] readily yields

f (a i α, b i β) = f * i (α, β) + O(P 1/4+ε
) uniformly for (α, β) ∈ W. When inserted into (8.2), the "trivial" bounds f (α, β) P (see (1.4)) and f * i (α, β) P (see (2.2)) suffice to infer that (8.5)

F (α, β) = s i=1 f * i (α, β) + O(P s-3/4+ε ).
We put (8.6) T (q) = q a,b=1 (a;b;q)=1 q -s s i=1 S(q, aa i , bb i ) and integrate (8.5) over W. Since the measure of W is O(P -k-3/8 ), this yields

W F (α, β) dα dβ = q≤P 1/8 T (q) P 1/8-k -P 1/8-k P -7/8 -P -7/8 s i=1 v(a i ξ, b i ζ) dζ dξ (8.7) + O(P s-k-9/8+ε ).
Note that the main term on the right hand side of (8.7) is a product. The next natural step is to complete the sum over q to a series, and likewise, to complete the integral to one extended over R 2 . This requires some care in cases where many of the b i are zero. In fact, when r = 3, the bounds in (2.2) are not of strength sufficient to conclude that the integrand in (8.7) is integrable over R 2 . In preparation for a debugging argument, let

v 0 (ξ, ζ) = 1 0 e(ξt k + ζt) dt.
By (2.1), one has v(ξ, ζ) = P v 0 (P k ξ, P ζ) so that an obvious substitution gives (8.8)

P 1/8-k -P 1/8-k P -7/8 -P -7/8 s i=1 v(a i ξ, b i ζ) dζ dξ = P s-k-1 J (P 1/8 ) where (8.9) J (W ) = W -W W -W s i=1 v 0 (a i ξ, b i ζ) dζ dξ. If ζ ≥
s i=1 v 0 (a i ξ, b i ζ) (1 + |ζ|) -2 (1 + |ξ|) -2 if |ζ| ≥ 8k|ξ|, (1 + |ξ|) -3 if |ζ| < 8k|ξ|.
To see this, first note that (7.3) (with

P = 1) yields v 0 (a i ξ, b i ζ) (1 + |ξ|) -1/k because |a i | ≥ 1.
Since s > 3k, this already shows the left hand side of (8.11) bounded by (1 + |ξ|) -3 irrespective of the value for ζ. If |ζ| ≥ 8k|ξ|, we estimate v 0 (a i ξ, b i ζ) as before for 3 ≤ i ≤ s, and this yields a factor (1 + |ξ|) -2 for the upper bound. Now recall that b 1 b 2 = 0, so that (8.10) gives v 0 (a

1 ξ, b 1 ζ)v 0 (a 2 ξ, b 2 ζ) (1 + |ζ|) -2 , as required.
The right hand side of (8.11) is an integrable function on R 2 , and it is immediate that its integral over max(|ξ|, |ζ|) ≥ W is O(W -1 ). It follows that (8.12)

J = J + a,b = ∞ -∞ ∞ -∞ s i=1 v 0 (a i ξ, b i ζ) dζ dξ
exists and that (8.13)

J (W ) = J + O(W -1
).

The treatment of the singular series also involves an unconventional element. We shall prove that (8.14)

T (q) q 2-s/k+ε .

The difficulty is implied by zero values of b i because then a factor S(q, aa i , 0) is present in (8.6), and when (b; q) = 1 but a = q, this factor is q and will not contribute to the savings needed to prove (8.14). As we shall see momentarily, some other factor in (8.6) will vanish whenever S(q, aa i , 0) is unduly large. To make this precise, we first apply the method leading from For p a, one may apply (2.2) to see that S(p l , aa i , bb i ) p l(1-1/k)+ε (even when b i = 0). This confirms (8.14) for prime powers, and by multiplicativity, (8.14) holds for all q. This estimate shows that the singular series (8.15)

S = S + a,b = ∞ q=1 T (q)
converges absolutely, and that q≤W

T (q) = S + O(W ε-1/k ).
We may now combine this with (8.7), (8.8) and (8.13) to conclude as follows.

Lemma 11. Let s ≥ 3k + 1 and r ≥ 3. Suppose that b is primitive. Then

W F (α, β) dα dβ = J + a,b S + a,b P s-k-1 + O(P s-k-1-1/(9k) ).
It remains to consider the complementary set of the major arcs W. Let N be the union of the intervals {α ∈ [0, 1] : |qα -a| ≤ P 4/5-k } with 0 ≤ a ≤ q, (a; q) = 1 and 1 ≤ q ≤ P 4/5 , and let n

= [0, 1] \ N. Now define V = N × [0, 1], v = n × [0, 1]. Note that [0, 1] 2 is the disjoint union of W, V \ W and v. For w ∈ {v, V \ W}, and 1 ≤ j ≤ 2, 3 ≤ i ≤ s, let Y ij (w) = w |f (a j α, b j β)| 2 |f (a i α, b i β)| s-2 dα dβ.
Then, applying the simple inequality

|f (a 1 α, b 1 β)f (a 2 α, b 2 β)| ≤ |f (a 1 α, b 1 β)| 2 + |f (a 2 α, b 2 β)| 2
together with Hölder's inequality, one infers that (8.16)

w |F (α, β)| dα dβ ≤ 2 j=1 s i=3 Y ij (w) 1/(s-2) .
We proceed by estimating Y ij (w). First consider the case where w = v, and 3 ≤ i ≤ s is an index with b i = 0. Then, by orthogonality, and recalling that b 1 b 2 = 0, one finds that

Y ij (v) = n |f (a i α, 0)| s-2 1 0 |f (a j α, b j β)| 2 dβ dα = [P ] n |f (a i α, 0)| s-2 dα.
/|a i | ∈ n} ∩ [0, 1] ⊂ m. Since |f (α, 0)| is of period 1, it follows that n |f (a i α, 0)| s-2 dα = 1 |a i | {α : α /|ai|∈n} |f (α , 0)| s-2 dα ≤ m |f (α, 0)| s-2 dα P s-2-k L -2 .
In the last step, we have used s ≥ 2 k + 2 and (1.5). By (8.17), this yields

(8.18) Y ij (v) P s-1-k L -2 .
Next consider the case where w = v and b i = 0. Then b i b j = 0, and Hölder's inequality gives

(8.19) Y ij (v) ≤ v |f (a j α, b j β)| s dα dβ 2/s v |f (a i α, b i β)| s dα dβ 1-(2/s)
.

Here, both integrals are of the same type. A substitution argument similar to the one in the preceding case yields

v |f (a j α, b j β)| s dα dβ = 1 |a j b j | {α : α/|ai|∈n} |bj | 0 |f (α, β)| s dβ dα ≤ m 1 0 |f (α, β)| s dβ dα P s-1-k L -2 .
Here, we used Theorem 2 for the last inequality. Because the same argument applies with i in place of j, we conclude that (8.18) holds in this case as well. Consequently, by (8.16),

v |F (α, β)| dα dβ P s-1-k L -2 . (8.20)
This leaves the set V \ W = U, say. A successful estimation is possible with the aid of Lemma 10. An inspection of the definitions of the sets V, W, K and L shows that whenever a j b j = 0, that

(α, β) : α/|a j |, β/|b j | ∈ U ∩ [0, 1] 2 ⊂ K \ L.
Hence, whenever b j = 0, one notes that f (α, β) is Z 2 -periodic to conclude that and Lemma 10 estimates the last integral as O(P s-k-1-δ ) where δ is any real number not exceeding 1/(9k). By using Hölder's inequality as in (8.19), it now follows that whenever j = 1, 2 and 3 ≤ j ≤ r, then (8.21) Y ij (U) P s-k-1-δ .

For r < i ≤ s, we merely use orthogonality, and use (1.5) together with a straightforward major arc estimate to infer that We now take w = U in (8.16). Since r ≥ 3, the bound (8.21) applies to at least one of the factors, so that one may deduce that We are ready to establish Theorem 1 in all cases where r ≥ 3. Let N (0) (P ) be the number of solutions counted by N a,b (P ) where x i = 0 for at least one index i with 1 ≤ i ≤ s. For the remaining solutions, we have x i = 0 for all i, and we write η i = x i /|x i |. We group the solutions according to a given value of η ∈ {1, -1} s , and substitute x i = η i x i in (1.1). We are then reduced to count solutions in positive integers of the system (1.1) with b i replaced by η i b i and a i replaced by η k i a i . Thus, on writing ηb = (η i b i ) 1≤i≤s , ηa = (η k i a i ) 1≤i≤s , we find that We shall establish the estimate (8.24) N (0) (P ) P s-k-1-δ at the end of this section. Taking this for granted, the asymptotic formula (1. However, substituting -x for x in (2.1) yields S(q, -a, -b) = S(q, a, b) when k is odd, and S(q, a, -b) = S(q, a, b) when k is even. By (8.6) and (8.15), this implies S + ηa,ηb = S + a,b = S. Moreover, since T (q) is multiplicative, we have Also, provided only that (1.1) has non-singular solutions in all Q p , we may apply Lemma 12 together with S = S + ηa,ηb , S > 0 and N + ηa,ηb (P ) ≥ 0 to conclude that J + ηa,ηb is real and non-negative, and in view of (8.28), the same is true for J. Now suppose that the equations i |. Then, the proof of [START_REF] Parsell | Pairs of additive equations of small degree[END_REF]Lemma 7.4] confirms that J + ηa,ηb > 0. Note that once this is established, it follows from (8.28) and the discussion following that formula that J > 0. Then, recalling the properties of S and C a,b = SJ, it follows that C a,b > 0 certainly holds whenever (1.1) admits non-singular solutions in all completions of Q. This establishes Theorem 1.

Y ij (U) ≤
We are left with the task to prove (8.24). First consider the contribution to N (0) (P ) that stems from solutions where two or more of the x i are zero. If, say, u ≥ 2 and x 1 = x 2 = • • • = x u = 0, x i = 0 for u < i ≤ s, then these solutions of (1.1) satisfy a u+1 x k u+1 + • • • + a s x k s = 0, and by writing their number as an integral suitable for application of the Hardy-Littlewood method, the inequalities of Hölder and Hua show that the number of these solutions does not exceed Lemma 5 coupled with Hölder's inequality shows that this is O(P s-k-1-δ ), for some δ > 0. Hence, the contribution to N (0) (P ) from solutions with x 1 = 0, x i = 0 (2 ≤ i ≤ s) is acceptable. The argument applies when some other variable vanishes (note that two b j with b j = 0 remain active, which is crucial), and this completes the proof of (8.24).

  ) N (P/d) -1 .

Theorem 2 .

 2 Let k ≥ 3. Then

  (α, β)| 2 k +2 dα dβ P 2 k -k+1 (log P ) -2 .Theorem 2 should be compared with the celebrated estimate(1.5) m |f (α, 0)| 2 k dα P 2 k -k (log P ) -2

e

  (ax k + bx)/q , v(ξ, ζ) = P 0 e(ξt k + ζt) dt.

Lemma 2 .

 2 Let p be a prime, and suppose that a, b are integers with p | a and p b. Then, for all l ∈ N, one has S(p l , a, b) = 0. Proof. On substituting x = p l-1 y + z in (2.1), one finds that S(p l , a, b) =

  the set of all n ≤ P with no prime factor in the interval [D, D ]. Then, uniformly for d ≤ P and m ∈ N, the number E d,m (P ) of n ∈ E with n ≡ m mod d satisfies (3.9) E d,m (P ) P log log P ϕ(d) log(2P/D) (see Lemma 2 of Vaughan [27] or Halberstam-Richert [12], Theorem 3.4). We now define the exponential sum (3.10) h(α, β) = x∈E e(αx k + βx).

  ,y)=0 |M(x,y)|>(P/D) k r(|M(x, y)|).

  y ≤ P }, let I(U ) denote the number of solutions counted by I that have (x, y) ∈ U . Recall the parameters D = L κ and D = P 1/6k . For 1 ≤ x ≤ P , let m(x) be the smallest prime factor of x that exceeds D if such a factor exists, and otherwise put m(x) = ∞. Consider the sets A = {(x, y) ∈ N 2 : x ≤ P, y ≤ P, (x; y) > D}, B = {(x, y) ∈ N 2 : x ≤ P, y ≤ P, (x; y) ≤ D, m(x) ≤ D }, C = {(x, y) ∈ N 2 : x ≤ P, y ≤ P, (x; y) ≤ D, m(y) ≤ D }. If a solution to (4.3) is counted by I, but the pair (x, y) is not in the union of A , B and C , then we have x ≤ P , y ≤ P , (x; y) ≤ D and m(x) > D , m(y) > D . In particular, x ∈ E and y ∈ E . Consequently I ≤ I(A ) + I(B) + I(C ) + I(E × E ). Moreover, by symmetry, I(B) = I(C ), and so (4.4) I ≤ 4 max I(X ) where X runs through the sets A , B and E × E . First suppose that I ≤ 4I(E × E ). Then recalling (3.10), orthogonality shows that

(4. 5 )

 5 f (α, β; W ) = w≤W e(αw k + βw)

4 0

 4 k + βy), g(α, β) = f (α, β, P/M ). k α, pβ)f p (-α, -β)|f (α, β)| t-2 dα dβ.One may reduce the estimation of I to bounding the integral (|f p (α, β)| K/2 |g(p k α, pβ)| K+2 dα dβ, which can be brought into play via Hölder's inequality. Indeed, on noting that |f p (-α, -β)| = |f p (α, β)| , one readily finds that

HM 2 (F

 2 3 , βh)| 2 |g(4α, β)| 8 dβ dα P/M ) 9/2 L η . By (6.10), (6.11) and Schwarz's inequality, (αh)E(αh 3 , βh)|g(4α, β)| 6 dβ dα P 17/4+3δ M -15/4 .

  r = 2 and (b 1 ; b 2 ) = 1, then any solutions to (8.1) has b 1 | x 2 , b 2 | x 1 . We substitute x 1 = b 2 y, x 2 = -b 1 y to infer that N (P ) equals the number of solutions to the equation (a 1 b k 2 -a 2 b k 1 )y k + a 3 x k 3 + . . . a s x k s = 0 with |x i | ≤ P and |y| ≤ P/ max(|b 1 |, |b 2 |). If a 1 b k 2 = a 2 b k 1

  (3.26) to(3.27) to confirm T (q) is multiplicative. Now let p be a prime, l ≥ 1 and suppose that p | a but p b. Since b is primitive, there is some b i with p b i , whence Lemma 2 gives S(p l , aa i , bb i ) = 0. By(8.6), it follows that T (p l ) = l , aa i , bb i ) because the remaining pairs a, b with (a; b; p) = 1 have p | a, and hence p b, and then the summand vanishes.

(8. 17 )

 17 We now substitute α = α|a i |. The definition of m in the preamble to Theorem 2 shows that {α : α

U

  |f (a j α, b j β)| s dα dβ = 1 |a j b j | {(α/|aj |,β/|bj |)∈U} |f (α, β)| s dα dβ ≤ K\L |f (α, β)| s dα dβ,

1 0 1 0 1 0

 111 |f (a i α, 0)| s-2 |f (a j α, b j β)| 2 dβ dα ≤ [P ] |f (a i α, 0)| s-2 dα P s-k-1 .

  β)| dα dβ P s-k-1-δ/s . An asymptotic formula for N + a,b (P ) is now available by combining the conclusions in (8.20) and(8.22) with those in Lemma 11. We summarise our work so far in the next lemma.Lemma 12. Let s ≥ 2 k + 2, r ≥ 3 and b be primitive. ThenN + a,b (P ) = J + a,b S + a,b P s-k-1 + O P s-k-1 L -2 .

(8. 23 )

 23 N a,b (P ) = N (0) (P ) + ηi∈{±1} N + ηa,ηb (P ).

8

 8 H , aa i , bb i = lim H→∞ p H(2-s) M a,b p H(8.27) where M a,b p H is the number of incongruent solutions to the system of congruencesa 1 x k 1 + • • • + a s x k s ≡ b 1 x 1 + • • • + b s x s ≡ 0 mod p H .In particular, it follows that χ p is real and non-negative. Because the product (8.26) converges absolutely, it now follows that there is a p 0 = p 0 (a, b) such thatS ≥ 1 2 p≤p0 χ p .Moreover, the method of proof of Parsell[START_REF] Parsell | Pairs of additive equations of small degree[END_REF] Lemma 7.5] combines with (8.27) to show that whenever (1.1) has a non-singular solution in Q p , then χ p > 0. Hence, if (1.1) has non-singular solutions in Q p for all primes p, then S > 0.We may now conclude that C a,b = SJ where (

(8. 29 ) a 1 ξ k 1 +

 291 . . . a s ξ k s = b 1 ξ 1 + • • • + b s ξ s = 0have a non-singular real solution ξ 0 . By the implicit function theorem, the equations (8.29) define a (s-2)-dimensional manifold in a neighbourhood of ξ 0 . Consequently, we may choose ξ 0 such that ξ (0) i = 0 for all 1 ≤ i ≤ s. By homogeneity, we may also suppose that |ξ(0) i | < 1, for 1 ≤ i ≤ s. Now define η i = ξ

1 0 1 0 1 0 1 0

 1111 |f (α, 0)| s-u dα P s-2-k+ε .By symmetry, this argument applies when any set of u variables x i vanishes.If exactly one of the variables vanishes, say x 1 = 0, then an analysis of the signs of the other variables similar to(8.23) reduces the problem of counting solutions to (1.1) withx 1 = 0 but 1 ≤ |x i | ≤ P (2 ≤ i ≤ s) to an estimate for i α, b i β)| dα dβ.Note that b 2 b 3 = 0 (recall r ≥ 3). An argument similar to (8.16) bounds the above integral byj α, b j β)| 2 |f (a i α, b i β)| s-3 dα dβ 1/(s-3).When b i = 0, we use orthogonality and the classical lemma of Hua to deduce that1 0 |f (a j α, b j β)| 2 |f (a i α, b i β)| s-3 dα dβ ≤ P |f (a i α, 0)| s-3 dα P s-k-1-δwhere δ > 0. When b i = 0, one may apply Hölder's inequality again, to separate f (a j α, b j β) from f (a i α, b i β). An obvious substitution then reduces the problem to that of estimating 1 0 |f (α, β)| s-1 dα dβ.

  |E(αh3 , βh)| 2 HM + M 2 H 3/4 (log H) 2 HM holds for M ≤ P 1/15 . To confirm these claims, let M < p 1 , p 2 ≤ 2M with p 1 = p 2 and |f α(p 6 1

	6 1 -p 6 2 ), β(p 1 -p 2 ), H	H 3/4 (log H) 2 .
	Once this is established, it follows that
	(6.6)

h≤H

  [START_REF] De La Bretèche | Répartition des points rationnels sur la cubique de Segre[END_REF]) is now available from(8.23) andLemma 12, with

	(8.25)	C(a, b) =	S + ηa,ηb J + ηa,ηb .
		ηi∈{±1}


Amelia Carolina Sparavigna

The Walled Town of Alife and the Solstices

Keywords: Italic Towns, Archaeology, Solstices, Saturnalia, Archaeoastronomy, SunCalc

Here we consider an astronomical orientation of the walled town of Alife, in Campania. This town has maintained its Roman original layout and the walls of that period. We will show, by means of SunCalc software, that the main street of Alife, its Decumanus, is aligned along the sunrise/sunset on solstices.

Introduction Alife is a town in the province of Caserta, Campania, located in the Volturno valley. The settlement had a Samnite origin. The Romans rebuilt it as an oppidum, Allifae, with the typical layout based on two mains roads at right angles, that divided it in four parts [START_REF] Vv | Alife, Wikipedia[END_REF]. This happened in 80-79 BC [START_REF] Parisi | L'acropoli e la cittadella: Murazioni sannitiche d'altura, borghi castramentati e castelli normanni intorno alla pianura alifana[END_REF]. The strong walls that are surrounding Alife are of the Roman time. Remains of an amphitheater are visible outside the walled town. The Roman Allifae is mentioned in [START_REF] Gall | Les romains et l'orientation solaire[END_REF]. This reference is considering the Roman centuriation, that is, the Roman land partitioning [START_REF] Sparavigna | Roman Centuriation in Satellite Images[END_REF] and the town-planning, and a possible solar orientation of their main axis -the Decumanus -along the direction of the sunrise. The author, Joël Le Gall, argued that there is no evidence of astronomical alignment of the Decumani. To sustain his conclusions, Le Gall considered some examples, and, as previously told, among the examples we find Allifae. « L'ancien bourg samnite d'Allifae a peut-être reçu une deductio à l'époque de Sylla. Nous ne connaissons guère que ses murailles, dont l'appareil en pseudo-réticulé paraît indiquer précisément elles dessinent un rectangle parfait à quatre portes axiales. Les deux directions étaient en dehors de l'amplitude. L'axe principal était de direction 2; cette direction était aussi celle de la Via Latina avec laquelle il se confondait: cette route étant antérieure, c'est elle qui avait dû la faire choisir pour la ville » [START_REF] Gall | Les romains et l'orientation solaire[END_REF]. About the town, the conclusion of Le Gall is that the Decumanus was aligned along the Via Latina, which was already existing. Giulio Magli, who argued against the thesis of Le Gall in his work on the astronomical orientation of the Roman towns in Italy [START_REF] Magli | On the orientation of Roman towns in Italy[END_REF], did not consider Allifae in the list of the 38 towns he analyzed for supporting "that astronomical orientation has been in many cases a fundamental component in the planning of Roman towns". Here we will show, using the satellite images and a software which is giving on them the direction of sunrise and sunset on any day of the year (SunCalc.net), that Alife has the Decumanus aligned along the sunrise/sunset on solstices. The fact that the direction of the Decumanus is the same of a preexisting Roman road does not exclude an astronomical orientation of the town.

Some further information on Allifae

In [START_REF] Parisi | L'acropoli e la cittadella: Murazioni sannitiche d'altura, borghi castramentati e castelli normanni intorno alla pianura alifana[END_REF] and in [START_REF] Allifae | The Princeton Encyclopedia of Classical Sites[END_REF], we find some further information about Allifae. The area was populated by the Samnite people of the Allibanos [START_REF] Parisi | L'acropoli e la cittadella: Murazioni sannitiche d'altura, borghi castramentati e castelli normanni intorno alla pianura alifana[END_REF]. The town in the 4th century BC was a part of Samnium; after the Augustan redistricting, it became part of Campania. In the first half of the 4th century, the city coined its own money [START_REF] Allifae | The Princeton Encyclopedia of Classical Sites[END_REF]. Later, the town was involved on many occasions in the problems of the Second and Third Samnite wars. Let us remember that the First, Second, and Third Samnite Wars (343-341 BC, 326-304 BC and 298-290 BC) were fought between the Roman Republic and the Samnites. Allifae was reduced to a colony during the second triumvirate (from 43 BC to 33 BC) [START_REF] Allifae | The Princeton Encyclopedia of Classical Sites[END_REF]. The territory of Allifae comprised a large area on the left of the Volturno, bounded by the Titerno and Lete rivers, and the Matese massif (Tifernus mons). Besides occasional finds from the prehistoric era, there is evidence of a settlement of the Iron Age in the area of Cila, above Piedimonte d'Alife. Many burial places exist over the entire area at the foot of the mountains, the most important of which are near Alife and Piedimonte, dating from the 7th century BC to the Roman era [START_REF] Allifae | The Princeton Encyclopedia of Classical Sites[END_REF]. References [START_REF] Parisi | L'acropoli e la cittadella: Murazioni sannitiche d'altura, borghi castramentati e castelli normanni intorno alla pianura alifana[END_REF][START_REF] Allifae | The Princeton Encyclopedia of Classical Sites[END_REF] tell us that the fortification wall at the foot of Mt. Cila were made by the Samnites. Of the Roman period, evidence supports the notion that the population was widely scattered or lived in fortified villages [START_REF] Parisi | L'acropoli e la cittadella: Murazioni sannitiche d'altura, borghi castramentati e castelli normanni intorno alla pianura alifana[END_REF]. Alife has a rectangular layout and still preserves a large part of its fortifications (Figure 1). The wall, in limestone opus incertum, with towers and four gates, apparently date to the Late Republican era [START_REF] Allifae | The Princeton Encyclopedia of Classical Sites[END_REF]. Of the same period was the theater. There are also some large funerary monuments from the period between the end of the Republic and the 1st century of the Empire. The amphitheater was placed outside the walls. As we made in some previous papers, let us use a software, the SunCalc.net, to evidence if the town has the Decumanus aligned along an astronomical direction. For instance, in [START_REF] Sparavigna | The Town Planning of Pompeii and Herculaneum Having Streets Aligned Along Sunrise on Summer Solstice[END_REF][START_REF] Sparavigna | Roman Towns Oriented to Sunrise and Sunset on Solstices[END_REF][START_REF] Sparavigna | Solstices at the Hardknott Roman Fort[END_REF][START_REF] Sparavigna | Wien and the Winter Solstice[END_REF][START_REF] Sparavigna | The Ancient Norba and the Solstices[END_REF], we have found that SunCalc (we can make the same using Sollumis.com or the Photographer's Ephemeris) gives the main street aligned along the direction of the sunrise on solstices. The software is easy to use: it shows the lines of sunrise and sunset on the satellite map of the chosen location on the desired day. Using SunCalc for Alife, we have the results given in the Figures 2 and3. The simulations in Figures 2 and3 tell us that the Decumanus is aligned along sunrise/sunset on solstices. Probably, in the rituals of foundation, it was the sunrise having the relevant role, so we can tell that Alife was founded with the Decumanus aligned along the sunrise on winter solstice.

Foundation ritual

In fact rituals for the land partition and for the foundation of a town existed. About such rituals, let us read some passages of a book written by Francis Haverfield on the ancient town planning [START_REF] John | was a British historian and archaeologist[END_REF][START_REF] Haverfield | Ancient Town-Planning[END_REF]. When Haverfield discusses the sites of Norba and Pompeii, two towns oriented according to solstices, he tells that evidence exists, which is relating the foundation rituals to "early Rome itself", and to ancient customs and observances. "These customs belong to the three fields of religion, agrarian land-settlement and war. All three exhibit the same principle, the division of a definite space by two straight lines crossing at right angles at its centre, and (if need be) the further division of such space by other lines parallel to the two main lines" [START_REF] Haverfield | Ancient Town-Planning[END_REF]. In this subdivision of the land, the "limitation", the Romans involved the augurs, the persons practicing augury, able of interpreting the will of the gods by studying the flight of birds. "The Roman augur who asked the will of Heaven marked off a square piece of sky or earth -his templum -into four quarters; in them he sought for his signs. The Roman general who encamped his troops, laid out their tents on a rectangular pattern governed by the same idea. The commissioners who assigned farming-plots on the public domains to emigrant citizens of Rome, planned these plots on the same rectangular scheme -as the map of rural Italy is witness to this day" [START_REF] Haverfield | Ancient Town-Planning[END_REF]. In fact, we can see this scheme in the satellite images of the Pianura Padana [START_REF] Sparavigna | Roman Centuriation in Satellite Images[END_REF]. Haverfield continues: "These Roman customs are very ancient. Later Romans deemed them as ancient as Rome itself, and, though such patriotic traditions belong rather to politics than to history, we find the actual customs well established when our knowledge first becomes full, about 200 BC. The Roman camp, for example, had reached its complex form long before the middle of the second century, when Polybius described it in words. Here, one can hardly doubt, are things older even than Rome. Scholars have talked, indeed, of a Greek origin or of an Etruscan origin, and the technical term for the Roman surveying instrument, groma, has been explained as the Greek word 'gnomon', borrowed through an Etruscan medium. But the name of a single instrument would not carry with it the origin of a whole art, even if this etymology were more certain than it actually is" [START_REF] Haverfield | Ancient Town-Planning[END_REF]. From these words, it is clear the Haverfield's opinion on the rituals concerning the limitation. They had origin in the Italic people, not necessarily from the Etruscan rituals. Haverfield explains that, when "the Roman antiquary Varro alleged that 'the Romans founded towns with Etruscan ritual', (oppida condebant Etrusco ritu), he set the fashion for many later assertions by Roman and modern writers. But he did not prove his allegation, and it is not so clear as is generally assumed, that he meant 'Etruscan ritual' to include architectural town-planning as well as religious ceremonial. These are Italian customs, far older than the beginnings of Greek influence on Rome, older than the systematic town-planning of the Greek lands, and older also than the Etruscans" [START_REF] Haverfield | Ancient Town-Planning[END_REF]. Let us note that, since the Etruscan ritual of land subdivision was based on the correspondence of the subdivision of the sky in four main parts [START_REF] Pallottino | The Etruscans[END_REF], and further into sixteen sectors, where each section was under the protection of a specific deity, this ritual seems not so different from Haverfield's ritual of Italic people concerning the land limitation. There was -Haverfield continues -an ancestral heritage of the Italian tribes, at the origin of customs and rituals. Probably, these customs were more secular rather than religious. "They took their rise as methods proved by primitive practice to be good methods for laying out land for farming or for encamping armies. But in early communities all customs that touched the State were quasi-religious; to ensure their due performance, they were carried out by religious officials. At Rome, therefore, more especially in early times, the augurs were concerned with the delimitation alike of farm-plots and of soldiers' tents. They testified that the settlement, whether rural or military, was duly made according to the ancestral customs sanctioned by the gods" [START_REF] Haverfield | Ancient Town-Planning[END_REF]. It may be asked how all this applies to the planning of towns. "We possess certainly no such clear evidence with respect to towns as with respect to divisions agrarian or military". But, as Haverfield observes, the Roman town-plans show very much the same outlines as those of the camp or of the farm plots. This outline of the urban planning, according to Haverfield, "it may well be called Italian". "We need not hesitate to put town and camp side by side, and to accept the statement that the Roman camp was a city in arms. … Whether the nomenclature of the augur, the soldier and the land-commissioner was adopted in the towns, is a more difficult, but fortunately a less important question. Modern writers speak of the cardo and the decumanus of Roman towns, and even apply to them more highly technical terms such as striga and scamnum. For the use of cardo in relation to towns there is some evidence. … The silence alike of literature and of inscriptions shows that they were, at the best, theoretical expressions, confined to the surveyor's office" [START_REF] Haverfield | Ancient Town-Planning[END_REF].

Saturn and the winter solstice For Alife, we can repeat what we have already mentioned for Norba [START_REF] Sparavigna | The Ancient Norba and the Solstices[END_REF]. In the framework discussed by Haverfield of an Italic ritual, we find an augur who was asking the will of Heaven by means of a space on earth -the templum -divided into four quarters (Alife is in fact clearly divided in quarters). For the orientation of the templum, it is possible he used, besides the local morphology of the site, also the astronomical phenomena, such as the rising of the sun. In the case of Alife, it seems that the augur linked the site to the winter solstice.

For Romans, the winter solstice was very important. The week about this solstice was a period of the great celebrations (Saturnalia) of the god Saturn, the agricultural deity who reigned over the world in the Golden Age. Saturn was also identified by the Romans to the Greek Kronos, the god of time [START_REF] Vv | [END_REF]. During the Saturnalia, the goodwill of this Age was restored for seven days. All business and military operations were postponed and people made exchange of friendly visits and gifts. Since Alife was built at the center of a fertile plane, it is probable that the orientation of the town along the sunrise on the winter solstice, was decided by the augurs as an homage to the god Saturn, who was the ruler of the Golden Age.

The town and the Roman road As we told in the introduction, when we mentioned the work of Le Gall, the fact that the direction of Alife's Decumanus is the same of a preexisting Roman road does not exclude an astronomical orientation of the town (and then of the road). As we can see by the map in Figure 4, roads are adapted to the local terrain. If we suppose that Via Alcide De Gasperi is following the ancient Via Latina, we have that it is bending to pass across Alife. The town is the center of the largest part of the plain shown by the map. It means that the town was also the center of a centuriation, or that the Roman road was the main road of a centuriation. If the town was adapted to a preexisting centuriation, it means that we could ascribe the observed astronomical orientation to this centuriation.

Figure 1 :

 1 Figure 1: Alife in Google Earth.

Figure 2 :

 2 Figure 2: SunCalc applied to Alife, on 21 December 2016.

Figure 3 :

 3 Figure 3: SunCalc applied to Alife, on 21 June 2016.

Figure 4 :

 4 Figure 4: Map of the roads (Courtesy HERE Map).