Quadratic forms and systems of forms in many variables

Let $F_1,\dotsc,F_R$ be quadratic forms with integer coefficients in $n$ variables. When $n\geq 9R$ and the variety $V(F_1,\dotsc,F_R)$ is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for $V(F_1,\dotsc,F_R)$. Previous work in this direction required $n$ to grow at least quadratically with $R$. We give a similar result for $R$ forms of degree $d$, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as $n>d2^dR$. In the case that $d\geq 3$, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients.

Let W be the projective variety cut out in P n−1 Q by the condition that the R × n Jacobian matrix (∂F i (x)/∂x j ) ij has rank less than R. If where the implicit constant depends only on the forms F i and δ is a positive constant depending only on d and R. If the variety V (F 1 , . . . , F R ) cut out in P n−1 Q by the forms F i has a smooth point over Q p for each prime p then S > 0, and if it has a smooth real point whose homogeneous co-ordinates lie in B then I > 0.
Here I, S are the usual singular integral and series; see (2.35) and (2.25) below.
We focus in particular on weakening the hypothesis (1.1) on the number of variables, when the number of forms R is greater than one. Previous improvements of this type have required R = 1 or 2. Our first result, proved in §4, is as follows: Theorem 1.2. When d = 2 and dim V (F 1 , . . . , F R ) = n−1−R, we may replace (1.1) with the condition n − σ R > 8R, (1.3) where σ R is the element of {0, . . . , n} defined by dim Sing V (β · F ), (1.4) and V (β·F ) is the of the hypersurface cut out in P n−1 R by β 1 F 1 +. . .+β R F R = 0.
Thus (1.3) is weaker than (1.1) whenever 2R(R + 1) < 8R holds, that is for R ≥ 4. To obtain the result described in the abstract we can simplify (1.3) with the following lemma, proved at the end of §4. Lemma 1.1. Let d ≥ 2 and let F 1 , . . . , F R and W be as in Theorem 1.1. If V (F 1 , . . . , F R ) is smooth with dimension n − 1 − R, then we have (1.6) If V (F 1 , . . . , F R ) is a smooth complete intersection and n ≥ 9R then Theorem 1.2 and Lemma 1.1 imply that the asymptotic formula (1.2) holds. This in turn implies that V (F 1 , . . . , F R ) satisfies the Hasse principle, by the last part of Theorem 1.1. As is usual with the circle method one also obtains weak approximation for V (F 1 , . . . , F R ) in this case; see the comments after the proof of Theorem 1.2 in §4.
The "square-root cancellation" heuristic discussed around formula (1.12) in Browning and Heath-Brown [7] suggests that the condition n > 4R should suffice in place of the n ≥ 9R in the previous paragraph. So (1.3) brings us within a constant factor of square-root cancellation as R grows, while (1.1) misses by a factor of O(R).
We deduce Theorem 1.2 from the following more general result, proved in §4.
Definition 1.1. For each k ∈ N \ {0} and t ∈ R k we write t ∞ = max i |t i | for the supremum norm. Let f (x) be any polynomial of degree d ≥ 2 with real coefficients in n variables x 1 , . . . , x n . For i = 1, . . . , n we define where we write x (j) for a vector of n variables (x Finally, for each B ≥ 1 we put N aux f (B) for the number of (d − 1)-tuples of integer n-vectors x (1) , . . . , x (d−1) with where we let for some C 0 ≥ 1, C > dR and all β ∈ R R and B ≥ 1, where we have written β · F for β 1 F 1 + · · · + β R F R . Then for all P ≥ 1 we have where the implicit constant depends at most on C 0 , C and the F i , and δ is a positive constant depending at most on C , d and R. Here I and S are as in Theorem 1.1.

One trivially has
So (1.8) requires us to save a factor of P 2 d C over the trivial upper bound, while the largest saving possible is of size O(P n ). It follows that we must have n > d2 d R in order for both (1.8) and C > dR to hold. Counting functions similar to N aux β·F (B) play a similar role in some other applications of the circle method, with the equations in place of the inequality (1.7). The quantities M (a 1 , . . . , a r ; H) from formula (9) of Dietmann [14], and M f (P ) from Lemma 2 of Schindler [30] are both of this type. In this setting one needs to save a factor of size B O(R 2 ) over the trivial bound.
In forthcoming work we bound the function N aux β·F (B) for degrees higher than 2, with the goal of handling systems F i in roughly d2 d R variables. We will approach this problem variously by using elementary methods, by generalising the argument used in Lemma 3 of Davenport [12] to treat the equations (1.9), and by applying the circle method iteratively to the inequalities (1.7). We will also combine the ideas used here with the variant of the circle method due to Freeman [15] to give a version of Theorem 1.3 for systems of forms F i with real coefficients.

Related work
Theorem 1 of Müller [27] gives a result with exactly the same number of variables as Theorem 1.2, but for quadratic inequalities with real coefficients rather than quadratic equations with rational coefficients. It is in turn founded on work of Bentkus and Götze [1,2] concerning a single quadratic inequality. The method of proof is related to ours, see § §2.1 and 3.1 below.
When d = 2, the forms F i are diagonal and the variety V (F 1 , . . . , F R ) is smooth, then the conclusions of Theorem 1.1 hold whenever n > 4R. That is, we have the "square-root cancellation" situation described at the end of §1.1. This follows by standard methods from a variant of Hua's lemma due to Cook [11]. When d = 2 Dietmann [13], improving work of Schmidt [31], gives conditions similar to (1.3) under which the asymptotic formula (1.2) holds and the constant S is positive. In particular it is sufficient that either min β∈C R \{0} rank(β · F ) > 2R 2 +3R, or that min a∈Q R \{0} rank(a·F ) > 2R 3 +τ (R)R, where τ (R) = 2 if R is odd and 0 otherwise. He also shows that if d = 2, the variety V (F 1 , . . . , F R ) has a smooth real point and min a∈Q R \{0} rank(a·F ) > 2R 3 −2R then V (F 1 , . . . , F R ) has a rational point.
Munshi [28] proves the asymptotic formula (1.2) when d = 2, n = 11 and V (F 1 , F 2 ) is smooth. By contrast using Theorem 1.1 and (1.6) would require n ≥ 14. When d = 2 and R = 1 we have a single quadratic form F . Heath-Brown [17] then proves such an asymptotic formula whenever V (F ) is smooth and n ≥ 3.
If F is a cubic form, Hooley [19] shows that when n = 8, the variety V (F ) is smooth, and B is a sufficiently small box centred at a point where the Hessian determinant of F is nonzero, then we have a smoothly weighted asymptotic formula analogous to (1.2). This result is conditional on a Riemann hypothesis for a certain modified Hasse-Weil L-function. For n = 9 he proves a similar result without any such assumption [18], with an error term O(P n−3 (log P ) −δ ) instead of the O(P n−3−δ ) in (1.2). In this setting Theorem 1.1 requires n ≥ 17.
In the case of a single quartic form F such that V (F ) is smooth, Hanselmann [16] gives the condition n ≥ 40 in place of the n ≥ 49 required to apply Theorem 1.1. Work in progress of Marmon and Vishe yields a further improvement.
When d ≥ 5 and R = 1, a sharper condition than (1.1) is available by work of Browning and Prendiville [8]. For d ≤ 10 and a smooth hypersurface V (F ) this is essentially a reduction of one quarter in the number of variables required.
Dietmann [14] and Schindler [30] show that the condition (1.1) may be re- (1.10) Note that the maximum here is over integer points, and so we may have σ Z < σ R .
Birch's work [3] is generalised to systems of forms with differing degrees by Browning and Heath-Brown [7] over Q and by Frei and Madritsch [?] over number fields. It is extended to linear spaces of solutions by Brandes [5,6]. Versions of the result for function fields are due to Lee [21] and to Browning and Vishe [9]. A version for bihomogeneous forms is due to Schindler [29], and Mignot [24,25] further develops these methods for certain trilinear forms and for hypersurfaces in toric varieties. Liu [22] proves existence of solutions in prime numbers to a quadratic equation in 10 or more variables. Asymptotic formulae for systems of equations of the same degree with prime values of the variables are considered by Cook and Magyar [10] and by Xiao and Yamagishi [32]. Magyar and Titchetrakun [23] extend these results to values of the variables with a bounded number of prime factors, while Yamagishi [33] treats systems of equations with differing degrees and prime variables. It is natural to ask whether similar generalisations exist for Theorem 1.2.

Notation
Parts of our work apply to polynomials with general real coefficients. Therefore we let f 1 (x), . . . , f R (x) be polynomials with real coefficients, of degree d ≥ 2 in n variables x 1 , . . . , x n , and we write f R (x) for the degree d parts. Implicit constants in ≪ and big-O notation are always permitted to depend on the polynomials f i , and hence on d, n, and R. We use scalar product notation to indicate linear combinations, so that for example (f ) and N aux f (B) are as in Definition 1.1. We do not require algebraic varieties to be irreducible, and we use the convention that dim ∅ = −1.
By an admissible box we mean a box in R n contained in the box [−1, 1] R , and having sides of length at most 1 which are parallel to the coordinate axes. We let B be an admissible box. For each α ∈ R R and P ≥ 1, we define the exponential sum S(α; P ) = where e(t) = e 2πit . This depends implicitly on B and the f i . We often write ∞ }, and if β = 0 this quantity is defined to be +∞.

Structure of this paper
In §2 we apply the circle method to a system of degree d polynomials with integer coefficients, assuming a certain hypothesis (2.1) on S(α; P ). In §3 we prove this hypothesis on S(α; P ) for polynomials with real coefficients, assuming that the bound (1.8) above holds. We then prove Theorems 1.2 and 1.3 in §4.

The circle method
In this section we apply the circle method, assuming that the bound min S(α; P ) P n+ǫ , holds for all α, β ∈ R R , P ≥ 1, some C > dR, C ≥ 1, and some small ǫ > 0. In particular we will show that (2.1) implies that the set of points α in R R where |S(α; P )| is large has small measure. Our goal is the result below, which will be proved in §2.5.
Proposition 2.1. Assume that the polynomials f i have integer coefficients, and that the leading forms f Suppose we are given C > dR, C ≥ 1 and ǫ > 0 such that the bound (2.1) holds for all α, β ∈ R R , all P ≥ 1 and all admissible boxes B. If ǫ is sufficiently small in terms of C , d and R, then we have N f1,...,fR (P ) = ISP n−dR + O C,f1,...,fR (P n−dR−δ ) for all P ≥ 1, all admissible boxes B, and some δ > 0 depending only on C , d, R. Here I, S are the usual singular integral and series given by (2.35) and (2.25) below.
We comment on the role of (2.1). If the f i have integer coefficients, then we have N f1,...,fR (P ) = If both S(α; P ) and S(α + β; P ) are large then (2.1) implies that one of the must be large. In particular, the points α and α + β must either be very close or somewhat far apart. In this sense (2.1) is a "repulsion principle" for the sum S(α; P ). We can use this fact to bound the measure of the set where S(α; P ) is large, and this will enable us to reduce (2.3) to an integral over major arcs.
To see the source of the condition C > dR in Proposition 2.1, consider the case In general we always have with equality when β ∞ = P 1−d holds. So in the case (2.4), the assumption (2.1) is trivial. In other words (2.1) might still be satisfied even if the function S(α; P ) had absolute value P n−C +ǫ at every point α in real R-space. This will lead to an error term of size at least P n−C +ǫ in evaluating the integral (2.3).
Hence we require C > dR in the proposition above in order for the error term to be smaller than the main term.

Mean values from bounds of the form (2.1)
We show that the bound (2.1) implies upper bounds for the integral of the function S(α; P ) over any bounded measurable set. Müller [27] and Bentkus and Götze [1,2] previously used similar ideas to treat quadratic forms with real coefficients.
We begin with a technical lemma. for the inverses of these maps. Let ν > 0 and let E 0 be a hypercube in R R whose sides are of length ν and parallel to the coordinate axes. Let E be a measurable subset of E 0 and let ϕ : E → [0, ∞) be a measurable function.
Suppose that for all α, β ∈ R R such that α ∈ E and α + β ∈ E, we have Then, for any integers k and ℓ with k < ℓ, we have where the implicit constant depends only on R.
Note that if we choose then the hypotheses (2.1) and (2.5) become identical. This will enable us to apply Lemma 2.1 to bound the integral m P,d,∆ S(α; P ) dα, where m P,d,∆ is a set of minor arcs on which S(α; P ) is somewhat small.
Proof. The strategy of proof is as follows. We deduce from (2.5) that if both ϕ(α) ≥ t and ϕ(α + β) ≥ t hold, then either β ∞ ≤ r 1 (t) or β ∞ ≥ r 2 (t) must hold. From this we will show that the set of points α satisfying the bound ϕ(α) ≥ t can be covered by a collection of hypercubes of side 2r 1 (t), each of which is separated from the others by a gap of size 1 2 r 2 (t). The lemma will follow upon bounding the total Lebesgue measure of this collection of hypercubes.
For each t > 0 we set Observe that if α and α + β both belong to D(t), then (2.5) implies that from which it follows that either β ∞ ≤ r 1 (t) or β ∞ ≥ r 2 (t) must hold. Let b be any hypercube in R R whose sides are of length 1 2 r 2 (t) and parallel to the coordinate axes. We claim that b ∩ D(t) is contained in a hypercube B whose sides are of length 2r 1 (t). To see this let α be any fixed vector lying in b ∩ D(t), and set In particular β ∞ < r 2 (t), so by the comments after (2.7), the bound β ∞ ≤ r 1 (t) must hold. This shows that α + β ∈ B, and hence that b ∩ D(t) ⊂ B, as claimed. In particular the Lebesgue measure of b ∩ D(t) is at most (2r 1 (t)) R .
The set D(t) is contained in E 0 , a hypercube of side ν. So in order to cover the set D(t) with boxes b of side 1 2 r 2 (t) one needs at most boxes. Summing over all the boxes b, it follows that With (2.8) this yields (2.6).
We now apply Lemma 2.1 to deduce mean values from bounds of the form (2.1). The following result is stated in greater generality than is strictly required here, to facilitate future applications to forms with real coefficients.
Lemma 2.2. Let T be a complex-valued measurable function on R R . Let E 0 be a hypercube in R R whose sides are of length ν and parallel to the coordinate axes, and let E be a measurable subset of E 0 . Suppose that the inequality holds for some P ≥ 1 and C > 0 and all α, β ∈ R R . Suppose further that (2.11) Later we will take T (α) = C −1 P −ǫ S(α; P ) where C is as in Proposition 2.1. We will take E to be a set of minor arcs m P,d,∆ , and we will interpret the integral m P,d,∆ S(α; P ) dα as an error term, which will need to be smaller than a main term of size around P n−dR . As a result, only the case C > dR of the bound (2.11) will be satisfactory for the present application.
Proof. We apply Lemma 2.1 with noting that the bound (2.5) then follows from (2.9). It remains to choose the parameters k and ℓ from (2.6). We will choose these so that the right-hand side of (2.6) is dominated by the sum ℓ−1 i=k , rather than either of the other two terms. More precisely, take observing that (2.14) We may assume that C > δ, for otherwise the bound E T (α) dα ≤ ν R P n−δ , which follows from (2.10), is stronger than any of the bounds listed in (2.11).
We then have k < ℓ and so this choice of k, ℓ is admissible in Lemma 2.1. Hence (2.6) holds, and substituting in our choices (2.12) for the parameters yields (2.15) By (2.10) and (2.14) we have sup α∈E |T (α)| P n ≤ 2 ℓ , and so we may extend the sum in (2.15) from ℓ−1 i=k to ℓ i=k to obtain Recall from (2.14) that we have 2 k ≥ 1 2 P −C and 2 ℓ ≤ 2P −δ , and observe that by (2.13) the bound ℓ − k ≤ 2 + C log 2 P holds. It follows that and reasoning similarly for with an implicit constant depending only on C , d, and R. One final application of the bound 2 k ≤ P −C from (2.14) completes the proof of (2.11).

Notation for the circle method
We split the domain [0, 1] R into two regions. Let ∆ ∈ (0, 1) and set We give local analogues of S(α; P ) and of the integral M P,d,∆ S(α; P ) dα. We set S q (a) = q −n y∈{1,...,q} n e a q · f (y) for each q ∈ N and a ∈ Z R , and we put For each γ ∈ R R , set and let Finally we define a quantity δ 0 which in some sense measure the extent to which the system f i is singular. Let σ Z ∈ {0, . . . , n} be as in (1.10), and let

The minor arcs
On the minor arcs m P,d,∆ we have the following bound, compare (2.10) in Lemma 2.2. Proof. The bound (2.18) follows either from Lemma 4 in Dietmann [14], or from Lemma 2.2 in Schindler [30], by setting the parameter θ in either author's work to be θ = ∆ − ǫ (d − 1)R , and taking P ≫ ǫ 1 sufficiently large. Provided the forms f

The major arcs
In this section we estimate M P,d,∆ S(α; P ) dα, the integral over the major arcs.
Lemma 2.4. Suppose that the polynomials f i have integer coefficients. Let ∆, M P,d,∆ , S ∞ (γ), S q (a), S(P ) and I(P ) be as in §2.2. Then for all a ∈ Z R and all q ∈ N such that q ≤ P , we have

19)
and it follows that Proof. To show (2.19) we follow the proof of Lemma 5.1 in Birch [3]. First If ψ is any differentiable complex-valued function on R n , then we have where the term q 1−n P n−1 allows for errors in approximating the boundary of the box B. Substituting into (2.21) shows that To complete the proof of (2.19) it suffices to set u = P t and use the definition of S ∞ (γ) from §2. We remark that in the case when a = 0 and q = 1, the proof of (2.19) is valid whether or not the polynomials f i have integer coefficients. That is, we always have for any f i with real cofficients. Next we treat the quantity S(P ) from (2.20).

Lemma 2.5.
Let the polynomials f i have integer coefficients, let the box B from §1.3 be [0, 1] n , and let S q (a) be as in §2.2. Suppose we are given ǫ ≥ 0 and C ≥ 1, such that for all α, β ∈ R R and all P ≥ 1 the bound (2.1) holds. Then: for all a ∈ {1, . . . , q} R and a ′ ∈ {1, . . . , q} R such that a ′ q ′ = a q .
(ii) If C > ǫ ′ , then for all t > 0 and q 0 ∈ N we have where it is understood that the fractions a q are in lowest terms.
(iii) Let δ 0 be as in §2.2 and let ǫ ′′ > 0. For all q ∈ N and all a ∈ Z R such that (a 1 , . . . , a R , q) = 1, we have (iv) Let ∆ and S(P ) be as in §2.2. Suppose that ǫ is sufficiently small in terms of C , d and R. Provided the inequality C > (d − 1)R holds and the forms f for some S ∈ C and some δ 1 > 0 depending at most on C , d and R. We have where the product is over primes p and converges absolutely.
Proof of part (i). Provided P is sufficiently large, Lemma 2.4 will allow us to approximate the sum S q (a) by a multiple of S a/q; P . This will enable us to transform (2.1) into the bound (2.23). Let P ≥ 1 be a parameter, to be chosen later. Then (2.1) gives min S a q ; P P n+ǫ , (2.26) Since B = [0, 1] n the equality S ∞ (0) = 1 holds, and so (2.19) implies that Observe that for P sufficiently large the term CP ǫ a ′ q ′ − a q C /(d−1) ∞ dominates the right-hand side of (2.28). We claim this is the case for Indeed, since a ′ q ′ − a q ∞ ≤ 1, it follows from (2.29) and (2.28) that which proves the result.
Proof of part (ii). If ǫ ′ < C is small, then by part (i), the points in the set such points fit in the box [0, 1) R , proving the claim.
Proof of part (iii). This follows from Lemma 2.3 by an argument which is now standard, see the proof of Lemma 5.4 in Birch [3].
Proof of part (iv). In this part of the proof, whenever we write a/q it is understood that a ∈ Z R and q ∈ N with (a 1 , . . . , a R , q) = 1. We will show below that for all Q ≥ 1, and some δ 1 > 0 depending only on C , d and R. Since where this sum is absolutely convergent. Then (2.25) follows as in §7 of Birch [3].
We prove (2.30). Let ℓ ∈ Z. We have Substituting these bounds into (2.32) gives We have C > (d − 1)R and we have assumed that ǫ ′ is small in terms of C , d and R, so we may assume that the bound C > (d − 1)R + ǫ ′ holds. So we may sum the geometric progression to find that Picking ℓ = ⌊log 2 Q δ0/2 ⌋ shows that .
The forms f (i) Suppose that the bound (2.1) holds for some C ≥ 1, C > 0 and ǫ ≥ 0 and all α, β ∈ R R and P ≥ 1. Then for all γ ∈ R R we have for some ǫ ′ > 0 such that ǫ ′ = O C (ǫ).
(ii) If the conclusion of part (i) holds and C − ǫ ′ > R, then there exists I ∈ C such that for all P ≥ 1 we have Furthermore we have Proof of part (i). First, for all β ∈ R R we have |S(β; P )| ≤ S(0; P ), from the definition (1.11). Consequently, taking α = 0, β = P −d γ in our hypothesis (2.1) shows that Together with the case α = P −d γ of the bound (2.22), this yields (2.36) If we have γ ∞ ≤ 1, then we set P = 1 and (i) follows at once. Otherwise we put P = max{1, γ 1+C ∞ }, and the result follows since (2.36) then implies Proof of part (ii). If the inequality C − ǫ ′ > R holds, then by (2.33) we have where the integrals converge absolutely. This proves (2.34) with (2.37) It remains to prove (2.35). Let χ : R R → [0, 1] be the indicator function of the box [− 1 2 , 1 2 ] R . We must evaluate the limit Let ϕ be any infinitely differentiable, compactly supported function on R R , taking values in [0, 1]. We evaluate R (t)) dt, which we think of as a smoothed version of (2.38). Fourier inversion gives Since C −ǫ ′ > R holds by assumption, it follows from (2.33) that the function S ∞ is Lebesgue integrable. Hence (2.37) implieŝ   With χ as in (2.38), choose ϕ such that ϕ(γ) ≤ χ(γ) for all γ ∈ R R . Then by (2.38) and (2.41) we have Letting ϕ → χ almost everywhere givesφ(0) → 1, so I is a lower bound for the limit inferior in (2.38). Repeating the argument with ϕ(γ) ≥ χ(γ) instead of ϕ(γ) ≤ χ(γ) shows that I is also an upper bound for the corresponding limit superior, so the limit exists and is equal to I.

The proof of Proposition 2.1
In this section we deduce Proposition 2.
With these choices for T , E 0 , E and δ we see that (2.9) follows from (2.1). Lemma 2.3 shows that sup α∈m P,d,∆ CT (α) ≪ ǫ P n−δ , and after increasing C if necessary this gives us (2.10). This verifies the hypotheses of Lemma 2.2. Since we have C > dR by assumption, (2.11) gives

The auxiliary inequality
In this section we verify the hypothesis (2.1), assuming a bound on the number of solutions to the auxiliary inequality from Definition 1.1. The goal is the following result, proved at the end of §3.2.
Proposition 3.1. Let N aux f (B), f ∞ be as in Definition 1.1. Suppose that we are given C 0 ≥ 1 and C > 0 such that for all β ∈ R R and B ≥ 1 we have

2)
noting that some such M, µ exist whenever the forms f are linearly independent. Let ǫ > 0. Then there exists C ≥ 1, depending only on C 0 , d, n, µ, M and ǫ, such that the bound (2.1) holds for all P ≥ 1 and all α, β ∈ R R .
Proof. Observe that (3.3) will follow if we can prove that S(α; P )S(α + β; P ) P 2(n+ǫ) First we use an idea from the proof of Theorem 5.1 in Bentkus and Götze [1], also found in Lemma 2.2 of Müller [26], to eliminate α. We have for some real polynomials g α,β,z (x) of degree at most d − 1 in x, and some boxes B z ⊂ B. Now by the special case of Cauchy's inequality | i∈I λ i | 2 ≤ (#I) · i∈I |λ i | 2 , we have (3.4) Bentkus and Götze used the double large sieve of Bombieri and Iwaniec [4] to bound the inner sum in (3.4) in the case when d = 2. We extend the argument to higher d by employing Lemma 2.4 of Birch [3], which states that 1 The innermost sum in (3.4) has the same form as S(α; P ), with B z in place of B and β · f [d] (x) + g α,β,z (x) in place of α · f as the underlying polynomial. The degree of g α,β,z is at most d − 1, so β · f [d] (x) is the leading part of this polynomial. So applying Birch's result to the innermost sum in (3.4) shows as U f depends only on the degree d part of f . With (3.4) this proves the result.
To show that S is positive under the conditions given in the theorem we use a variant of Hensel's Lemma. Let p be a prime and let a ∈ Z n p . Suppose that x = a is a solution to the system f i (x) = 0 for which the Jacobian matrix (∂f i (x)/∂x j ) ij is nonsingular. Possibly after permuting the variables x i if necessary, we can assume that the submatrix M (x) consisting of the last R o( β (i) · F ∞ ). By passing to a subsequence, we can assume β (i) / β (i) ∞ → β, and then at least σ R + 1 of the eigenvalues of M (β · F ) must be zero. In other words, dim Sing V (β · F ) ≥ σ R .
As alluded to after Lemma 1.1, the argument used to prove Theorems 1.2 and 1.3 also yields weak approximation for V (F 1 , . . . , F R ) if that variety is smooth. It suffices to show that if the system F i (qx − a) = 0 has solutions in the p-adic integers for each p, then it has integral solutions x with x x ∞ arbitrarily close to r r ∞ , for any fixed real solution r to the system F i (r) = 0. For this one can let B be a sufficiently small box containing r r ∞ , and repeat the proof of Theorems 1.2 and 1.3 with the choice f i (x) = F i (qx − a) instead of f i = F i at the start of the proof of Theorem 1.3. Since N aux β·f (B) = N aux β·F (B) we obtain (4.1) as before. Recalling that any real or p-adic point of V (F 1 , . . . , F R ) must be smooth, the argument to prove that I, S are positive goes through and we obtain the existence of an integral solution of the required kind.
Since V (F 1 , . . . , F R−1 ) has dimension n − 1 − R, it follows that V (F 1 , . . . , F R−1 ) ∩ Sing V (β · F ) ⊂ Sing V (F 1 , . . . , F R ) and so V (F 1 , . . . , F R−1 ) ∩ Sing V (β · F ) = ∅, as V (F 1 , . . . , F R ) is smooth. It follows that dim Sing V (β ·F ) ≤ R − 1, which proves the first inequality in (1.6). The second inequality in (1.6) follows from the work of Browning and Heath-Brown [7]. In those authors' formula (1.3), set D = 2, r 1 = 0, r 2 = R, F i,2 = F i . Now the R × n Jacobian matrix (∂F i (x)/∂x j ) ij has full rank at every nonzero solution x ∈ Q n to F 1 (x) = · · · = F R (x) = 0, because V (F 1 , . . . , F R ) is smooth of dimension n − 1 − R. This makes F i,j a 'nonsingular system" in the sense of Browning and Heath-Brown, as defined in their formula (1.7). The next step is to replace F i,d with an "equivalent optimal system". The comments after formula (1.7) of those authors show that in our case this means replacing F i with j A ij f j , where A is an invertible linear transformation. In particular this preserves V (F 1 , . . . , F R ) and W . Now their formulae (1.4) and (1.8) show that B 2 ≤ R − 1, where B 2 = 1 + dim(W ). This proves (1.6).