Rational points and prime values of polynomials in moderately many variables

Bull. Sci. math. 156 (2019) 102794

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Bulletin des Sciences Mathématiques www.elsevier.com/locate/bulsci

Rational points and prime values of polynomials in moderately many variables Kevin Destagnol a , Efthymios Sofos b,∗ a b

IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn, 53111, Germany

a r t i c l e

i n f o

Article history: Received 21 September 2018 Available online 14 August 2019 MSC: 11N32 11P55 14G05 Keywords: Rational points Schinzel’s hypothesis

a b s t r a c t We derive the Hasse principle and weak approximation for fibrations of certain varieties in the spirit of work by ColliotThélène–Sansuc and Harpaz–Skorobogatov–Wittenberg. Our varieties are defined through polynomials in many variables and part of our work is devoted to establishing Schinzel’s hypothesis for polynomials of this kind. This last part is achieved by using arguments behind Birch’s well-known result regarding the Hasse principle for complete intersections with the notable difference that we prove our result in 50% fewer variables than in the classical Birch setting. We also study the problem of square-free values of an integer polynomial with 66.6% fewer variables than in the Birch setting. © 2019 Elsevier Masson SAS. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (K. Destagnol), [email protected] (E. Sofos). https://doi.org/10.1016/j.bulsci.2019.102794 0007-4497/© 2019 Elsevier Masson SAS. All rights reserved.

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1. Introduction 1.1. Prime values of polynomials and rational points Let n  1 be an integer and assume that f ∈ Q[t1 , . . . , tn ]. Let K1 , . . . , Kr be cyclic extensions of Q, denote the degree [Ki : Q] by di and fix a basis {ω1,i , . . . , ωdi ,i } for Ki as a vector space over Q. We let NKi /Q be the field norm and denote NKi /Q (xi ) = NKi /Q (x1,i ω1,i + · · · + xdi ,i ωdi ,i ),

(1  i  r).

Let now the quasi-affine variety X ⊂ An × Ad1 × · · · × Adr be defined via   X : 0 = f (t1 , . . . , tn ) = NK1 /Q (x1 ) = · · · = NKr /Q (xr )

(1.1)

and let V be a smooth proper model of the affine subvariety of An × Ad1 × · · · × Adr given by f (t1 , . . . , tn ) = NK1 /Q (x1 ) = · · · = NKr /Q (xr ).

(1.2)

The Hasse principle and weak approximation for varieties of this kind have been the object of intensive study. There are cases where the Hasse principle and weak approximation hold and there are examples for which they fail, [9]. However, it has been conjectured by Colliot-Thélène [8] that all such failures are accounted for by the Brauer–Manin obstruction. The main objective of this paper is to study the Hasse principle and weak approximation for the class of varieties defined by (1.1) and (1.2) under the restriction that the polynomial f is an irreducible form and has many variables compared to its degree, but only moderately so as it will appear in due course. To this end, information on prime values assumed by integer polynomials can be exploited. The prototypical example is due to Hasse [21], whose proof of the Hasse principle for smooth quadratic forms in four variables relies on Dirichlet’s theorem on primes in arithmetic progressions combined with the global reciprocity law and the Hasse principle for non-singular quadratic forms in three variables. This fibration argument was later generalized in an important work by Colliot-Thélène and Sansuc [9] to establish that, conditionally under Schinzel’s hypothesis, various pencils of varieties over Q satisfy the Hasse principle and weak approximation. Their result was then extended by many authors, see the introduction of [20] for a list of relevant references. Theorem 1.3 below will allow us to replace Schinzel’s hypothesis in order to prove unconditionally the Hasse principle and weak approximation for the varieties defined by (1.1) and (1.2) in the case of an irreducible form f with moderately many variables compared to its degree. Let us conclude by mentioning that unconditional proofs in this subject exist in cases where the underlying polynomials have small degree (see for instance [10, Th. 9.3]) or special factorisation over Q. For example, polynomials

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that are completely split over Q are treated in [5]. Our main result provides an example where the polynomial needs to have moderately many variables compared to its degree but has no restriction on its shape. For a homogeneous polynomial f ∈ Z[x1 , . . . , xn ] that is irreducible in Q[x1 , . . . , xn ] we let σf be the dimension of the singular locus of f = 0, namely the dimension of the affine variety cut out by the system of equations ∇f (x1 , . . . , xn ) = 0 (see [2, pg. 250]). Observe that 0  σf  n − 1 and that σf = 0 if and only if the projective variety defined by f is non-singular. Theorem 1.1. Let f, Ki and X be as in (1.1) with f an irreducible form and assume that n − σf  max{4, 1 + 2deg(f )−1 (deg(f ) − 1)}. Then X satisfies the Hasse principle and weak approximation. In particular, X(Q) is Zariski dense as soon as it is non-empty. Note that, thanks to the fact that our Theorem 1.3 below (which is the main ingredient of the proof of Theorem 1.1) holds in at least half as many variables as in the work of Birch, a direct application of [2, §7, Th. 1] would not prove Theorem 1.1. Our strategy will be to establish an analogue of [20, Prop. 1.2] and then to adapt the argument in the proof of [20, Th. 1.3]. Like in the proof of [10, Th. 9.3], one can deduce weak approximation for the variety V defined by (1.2) from Theorem 1.1, since weak approximation is a birational invariant of smooth varieties and hence it is enough to establish the result for the smooth model of V provided by X. We then conclude the proof of Corollary 1.2 by alluding to the fact that the Hasse principle and Zariski density by the existence of a rational point are consequences of weak approximation. Corollary 1.2. Keep the assumptions of Theorem 1.1 and let V be as in (1.2). Then V satisfies the Hasse principle and weak approximation. In particular, V (Q) is Zariski dense as soon as it is non-empty. 1.2. Primes represented by polynomials As mentioned above, a key tool in our proof of Theorem 1.1 is a generalization of Schinzel’s hypothesis for polynomials in moderately many variables compared to its degree. Let f ∈ Z[x1 , . . . , xn ] be a, not necessarily homogeneous, polynomial that is irreducible in Q[x1 , . . . , xn ] and denote by f0 the top degree part of f . We make the definition σf := σf0 . For a non-empty compact box B ⊂ Rn with the property that f0 (B ) ⊂ (1, ∞), we define  dx . (1.3) πf (B ) := #{x ∈ Zn ∩ B : f (x) is a prime} and Lif (B ) := log f0 (x) B

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Our main result in this section is the following theorem. Theorem 1.3. Assume that f ∈ Z[x1 , . . . , xn ] is any integer polynomial which is irreducible in Q[x1 , . . . , xn ] and let B ⊂ Rn be any non-empty compact box having the property f0 (B ) ⊂ (0, ∞). If n − σf  max{4, (deg(f ) − 1)2deg(f )−1 + 1},

(1.4)

then for every fixed A > 0 the following holds for all sufficiently large P ,  πf (P B ) =



p prime

(1 − p−n #{x ∈ Fpn : f (x) = 0}) (1 − 1/p)



 Lif (P B ) + OA,B,f

Pn (log P )A



where the implied constant depends at most on A, B and f . Note that the assumption f0 (B ) ⊂ (0, ∞) shows that for all sufficiently large P every x ∈ P B satisfies f0 (x)  P deg(f ) > 1, thus Lif (P B ) is well-defined. We shall see in Lemma 3.22 that Lif (P B ) − vol(B )P n / log(P deg(f ) ) f,B P n /(log P )2 , hence vol(B ) πf (P B ) log P − Pn deg(f )



 (1 − p−n #{x ∈ Fpn : f (x) = 0}) (1 − 1/p) p

 f,B

1 . log P

(1.5)

Bateman and Horn [1] provided heuristics that led to a conjecture regarding the prime values of an integer polynomial in a single variable. One can modify their heuristics in the case that the polynomial has arbitrarily many variables, thus resulting in an analogous conjecture regarding the prime values of an integer polynomial in many variables. We refer the reader to Appendix A for a quick overview of Schinzel’s hypothesis, the Bateman–Horn conjecture and their generalisations. Theorem 1.3 then establishes the analogous conjecture provided that the polynomial has sufficiently many variables compared to its degree. There are currently no available techniques capable of settling any case of the Bateman–Horn conjeture in one variable apart from the case of one linear polynomial, which is Dirichlet’s theorem for primes in arithmetic progressions. Efforts have therefore focused on settling such problems for polynomials in more variables. Notable examples in cases with n = 2 are Iwaniec’s work [25] for quadratic polynomials, Fouvry–Iwaniec’s work [15] for x21 + x22 with x2 prime, Friedlander–Iwaniec’s work [16] for x21 + x42 , HeathBrown’s work [22] for x31 + 2x32 , Heath-Brown–Moroz’s work [24] for binary cubic forms and the recent work of Heath-Brown–Li [23] on x21 + x42 with x2 prime. The special shape of these polynomials plays a central rôle in the proofs of these results; they are all related to norms of a number field. In cases with n > 2 it should be noted that Green–Tao– Ziegler [19] studied simultaneous prime values of certain linear polynomials by a variety of methods, Friedlander and Iwaniec [17] studied the prime values of x21 + x22 + x23 via the

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class number formula of Gauss, while Maynard’s work [28] employs geometry of numbers to cover the case of incomplete norm forms. It is therefore a natural question whether the problem of representing primes by polynomials can be studied for polynomials with no special shape. Let us recall here that one of the important theorems in the frontiers between analytic number theory and Diophantine geometry concerns the Hasse principle for systems of polynomials in many variables and with no special shape by Birch [2]. To prove Theorem 1.3 we shall employ the Hardy–Littlewood circle method in the form used by Birch and use several of his estimates. While Birch’s work applies to every non-singular homogeneous polynomial f having at least n  (deg(f ) − 1)2deg(f ) + 1 variables (which was recently improved by Browning

and Prendiville [6] to n  (deg(f ) − deg(f )/2)2deg(f ) ), the assumption (1.4) of our Theorem 1.3 is less restrictive, as it allows for half as many variables. The improved range is due to the use of L2 -norm inequalities in the minor arcs, as well as bounds for exponential sums due to Browning–Heath-Brown [4] and Deligne [13] to show that the singular series in Birch’s work converges absolutely in the range (1.4). Let us finally give a direct consequence of Theorem 1.3. Corollary 1.4. Let f ∈ Z[x1 , . . . , xn ] be an integer homogeneous polynomial which is irreducible in Q[x1 , . . . , xn ] and assume that n − σf  max{4, (deg(f ) − 1)2deg(f )−1 + 1}. Then f (x) takes infinitely many distinct positive prime values as x ranges over Zn if and only if f (Rn ) is not included in (−∞, 0] and for every prime p the set f (Zn ) is not included in pZ. Proof. We clearly need to focus only on the sufficiency. If f (Rn ) ⊂ (−∞, 0] holds, then we can obviously find a non-empty box B ⊂ Rn with f (B ) ⊂ (0, +∞), so that vol(B ) = 0. If f (Zn ) ⊂ pZ holds, then the p-adic factor in (1.5) is strictly positive and we shall see in Lemma 3.17 that the product over p is absolutely convergent. Hence by (1.5) we deduce that πf (P ) f,B P n / log P . If f (x) = q was soluble only for finitely many primes q, say q1 , . . . , qr , then the standard estimate #{x ∈ Zn ∩ P B : f (x) = q} f,q,B P n−1 would lead to  Pn

f,B πf (B) = #{x ∈ Zn ∩ P B : f (x) = q} f,q1 ,...,qr ,B P n−1 , log P i=1 r

which is a contradiction. 2 1.3. Square-free integers represented by polynomials An integer m is called square-free if for every prime p we have p2  m. In particular, 0 is not square-free and m is square-free if and only if −m is.

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Assume that we are given a polynomial f ∈ Z[x1 , . . . , xn ] that is separable as an element of Q[x1 , . . . , xn ] and let f0 and σf be as in §1.2. A similar approach to the one for Theorem 1.3 allows us to study the set Sf := {x ∈ Zn : f (x) is square-free}. Theorem 1.5. Assume that f ∈ Z[x1 , . . . , xn ] is any integer polynomial which is separable as an element of Q[x1 , . . . , xn ] and let B ⊂ Rn be any non-empty closed box. If 1

n − σf > max 1, (deg(f ) − 1)2deg(f ) , 3

(1.6)

then there exists β = β(f ) > 0 such that for all P  2 the equality     #{Sf ∩ P B } −2n 2 n = # x ∈ (Z/p Z) : f (x) = 0 + Of,B (P −β ) 1 − p #{Zn ∩ P B } p prime holds with an implied constant that depends at most on f and B . The problem of square-free values of integer polynomials has a very long history, see [3] for a list of references. Many cases are still open, for example, there is no irreducible quartic integer polynomial in one variable for which we know that it takes infinitely many square-free values. One of the most general results, conditional on the abc conjecture, is due to Poonen [29] where arbitrary polynomials are treated. Our Theorem 1.5 covers unconditionally arbitrary polynomials of fixed degree and number of variables with the proviso that the number of variables is suitably large compared to the degree. Theorem 1.5 features a saving of two thirds of the variables compared to the Birch setting [2]. This saving comes from the fact that exponential sums whose terms are restricted to square-free integers can be bounded in a satisfactory manner, this was done in the work of Brüdern, Granville, Perelli, Vaughan and Wooley [7] and Keil [27]. Notation. We shall use the notation x to refer to n-tuples x = (x1 , ..., xn ). We will also make use of the classical von Mangoldt function denoted Λ and of the classical Möbius function denoted μ. The letter d will refer exclusively to the degree of the polynomial f in Theorem 1.3. Finally, throughout the paper, we shall make use of the notation e(z) := exp(2πiz), z ∈ C.

(1.7)

The polynomial f and the box B will be considered fixed throughout. This is taken to mean that, although each implied constant in the big O notation will depend on several quantities related to f , we shall avoid recording these dependencies. The list of the said quantities consists of f0 , n, d, σf , θ0 , δ, η, λ1 , A, λ,

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whose meaning will become evident in due course. The symbol ε will be used for a small positive parameter whose value may vary, allowing, for example, inequalities of the form xε xε/4 . Further dependency of the implied constants on other quantities will be recorded explicitly via an appropriate use of subscript. Acknowledgements. We are grateful to Jean-Louis Colliot-Thélène and Yang Cao for helpful conversations regarding the applications of Theorem 1.3. We would also like to thank Tim Browning for suggesting the proof of Proposition 3.7. We wish to acknowledge the comments of the anonymous referee that helped improve this work considerably. 2. The proof of Theorem 1.1 Denote by Qv the completion of Q with respect to the place v, let | · |p be the p-adic norm defined by |x|p = p−νp (x) for x ∈ Qp if v = p is finite and define | · |∞ as the classical absolute value for the real place. We will use the notation ZS for the ring of S-integers for any finite set of finite places S and we will say that a prime p is a fixed prime divisor of a polynomial f ∈ Z[x1 , . . . , xn ] if, for all (x1 , . . . , xn ) ∈ Zn , we have p | f (x1 , . . . , xn ). 2.1. Preliminary lemmas We begin by establishing the following analogue of [9, Lem. 2]. Lemma 2.1. Let f ∈ Z[x1 , . . . , xn ] be a non-zero polynomial with content equal to 1. If p is such that f (Zn ) ⊆ pZ, then p  deg(f ). Proof. Define d := deg(f ) and let p be a prime such that f (Zn ) ⊆ pZ. On one hand, we have by assumption that n

# {x ∈ (Z/pZ)

: f (x) = 0} = pn .

On the other hand, since f has content one, [33, Eq. (2.7)] implies that n

# {x ∈ (Z/pZ)

: f (x) = 0}  dpn−1

and hence p  d, thus concluding the proof of the lemma. 2 We now use Lemma 2.1 to verify the following analogue of [20, Prop. 1.2] and of the hypothesis (H1 ) of [11] over Q. Proposition 2.2. Let f ∈ Q[x1 , . . . , xn ] be an irreducible homogeneous polynomial satisfying the assumptions (1.4) and f (1, 0, . . . , 0) > 0. Let C be a positive real constant and ε > 0. Suppose we are given (λ1,p , . . . , λn,p ) ∈ Qnp for p in a finite set of finite

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places S containing all primes p  deg(f ) and all primes p such that f does not have p-integral coefficients as well as all primes p such that νp (f (1, 0, . . . , 0)) > 0. Then there exists infinitely many (λ1 , . . . , λn ) ∈ ZnS such that λ1 > Cλi > 0 for all i ∈ {2, . . . , n}, |λi − λi,p |p < ε for all i ∈ {1, . . . , n} and p ∈ S and f (λ1 , . . . , λn ) = u for a prime ∈ /S × and u ∈ ZS , u > 0. Proof. Up to multiplication of (λ1 , . . . , λn ) and (λ1,p , . . . , λn,p ) by a product of powers of primes in S, we can assume without loss of generality that (λ1,p , . . . , λn,p ) ∈ Znp for p ∈ S. The assumption that f (1, 0, . . . , 0) > 0 provides with ai ∈ Q and a > 0 such that 

f (x1 , . . . , xn ) = axd1 +

ai xi11 · · · xinn .

i=(i1 ,...,in )∈N i1 +···+in =d 0i1 ,...,in d i1 =d

(2.1)

n

Let N<0 be the number of indices i with ai < 0. We can assume C > N>0a|ai | and C > 1 whenever ai < 0. As in the proof of [20, Prop. 1.2], we can now find (λ0,1 , . . . , λ0,n ) ∈ Zn such that |λ0,i −λi,p |p < ε/2 for all p ∈ S. We can choose them such that λ0,1 > Cλ0,i > 0 for all i ∈ {2, . . . , n}. We can now see that f (λ0,1 , . . . , λ0,n ) > 0 by alluding to f (λ0,1 , . . . , λ0,n )  aλd0,1 −



1 n |ai |λi0,1 · · · λi0,n ,

ai <0

and the inequalities a  d λ N>0 a <0 0,1 i  a  a  i1 n 1 n 1 n = λ0,1 · · · λi0,1 > C d−i1 λi0,1 · · · λi0,n > |ai |λi0,1 · · · λi0,n . N>0 a <0 N >0 a <0 a <0

aλd0,1 =

i

i

i



Let A = p∈S p and a fixed integer N big enough so that |AN |p < ε/2 for all p ∈ S. Now consider the polynomial g ∈ Q[x1 , . . . , xn ] given by g(x1 , . . . , xn ) = f (λ0,1 + x1 AN , . . . , λ0,n + xn AN ). The polynomial g can be expressed as g = t˜ g for t ∈ Z× ˜ a polynomial with integer S and g coefficients which is irreducible over Q. Let us denote by c the product of all fixed prime factors of g˜. We will now establish that if p is a prime factor of c then p ∈ S. Let p | c. Either p divides the content of g˜ and in particular, with the notation (2.1), νp (aA) = 0 which immediately implies that p ∈ S or, denoting by c˜ the content of g˜, p is a fixed prime factor of the polynomial g˜/˜ c which has integral coefficients and content equal to one. By Lemma 2.1 this implies that p  deg(f ) and hence that p ∈ S. Moreover, with the notation of §1.2, g˜0 = AdN f and the conditions x1 > Cxi > 0 define an open cone C in Rn . In addition, when f is evaluated at (λ0,1 , . . . , λ0,n ) ∈ C it produces a strictly

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positive value, therefore we can find a box B ⊆ C such that f (B ) ⊂ (0, ∞). Since for all P we have P B ⊂ C , we obtain from Theorem 1.3 that there exist infinitely many x ∈ Zn ∩ C such that g˜(x)/c is prime. Introducing λi = λ0,i +xi AN for any i ∈ {1, . . . , n} and for any such x ∈ Zn ∩ C , we get that f (λ1 , . . . , λn )/(ct) = g(x1 , . . . , xn )/(ct) is prime. This yields the result because λ0,1 > Cλ0,i > 0 and x1 > Cxi > 0, which implies λ1 = (λ0,1 + x1 AN ) > Cλi = C(λ0,i + xi AN ). Moreover, |λi − λi,0 |p  |AN |p < ε/2 and hence |λi − λi,p |p < ε for all p ∈ S and all i ∈ {1, . . . , n}. 2 2.2. Conclusion of the proof of Theorem 1.1 Proof. We proceed by adapting the proof of [20, Th. 1.3]. We are given 1 > ε > 0, a finite set of places S and a point (tv , x1,v , . . . , xr,v ) ∈ X(Qv ) for every place v of Q and we want to find (t, x1 , . . . , xr ) ∈ X(Q) such that for all v ∈ S, 

|ti − ti,p |v < ε

(i ∈ {1, . . . , n}),

|xji ,i − xji ,i,v |v < ε

(i ∈ {1, . . . , r},

ji ∈ {1, . . . , di }).

2.2.1. First step By density and continuity, we can assume that t∞ ∈ Qn and by a linear change of variables, we can assume that t∞ = (1, 0, . . . , 0). Note that the solubility over R implies that f (1, 0, . . . , 0) > 0 in the case where there is a totally imaginary Ki . In addition, it implies that f (1, 0, . . . , 0) can be strictly positive or strictly negative when all Ki are totally real. We denote by s ∈ {−1, +1} the sign of f (1, 0, . . . , 0). We can enlarge S so that it contains the real place, the field Ki is unramified outside S for all i ∈ {1, . . . , r}, S contains all primes p  deg(f ), all primes p such that f does not have p-integral coefficients as well as primes p such that νp (f (1, 0, . . . , 0)) > 0. 2.2.2. Second step Let L = [d1 , . . . , dr ] denote the least common multiple of the degrees d1 , . . . , dr and M = maxv∈S max 1ir |xji ,i,v |v . By [14, Prop. 6.1], we know that the image of the 1ji di

× map NKi,p /Qp : Ki,p → Q× p is open and that a polynomial function is continuous for the p-adic topology. Hence there exists ε > 0 such that for every (λ1 , . . . , λn ) ∈ Qn satisfying for every i ∈ {1, . . . , n}, the inequality |λi − ti,p |p < ε , we have that f (λ1 , . . . , λn ) is a local norm for Ki /Q for the place p. Furthermore, there exists ε > 0 such that for every (λ1 , . . . , λn ) ∈ Qn satisfying for every i ∈ {1, . . . , n}, the inequality |λi − ti,p |p < ε , we have that

|f (λ1 , . . . , λn ) − f (t1,p , . . . , tn,p )|p <

ε |f (t1,p , . . . , tn,p )|p |L|p , 2M

(p ∈ S).

(2.2)

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Then applying Proposition 2.2 yields (λ1 , . . . , λn ) ∈ ZnS such that λ1 > Cλi > 0 for all i ∈ {2, . . . , n}, |λi − ti,p |p < min{ε/2, ε , ε /2} for all i ∈ {1, . . . , n} and p ∈ S and f (λ1 , . . . , λn ) = s u for a prime ∈ / S and u ∈ Z× S with u > 0. We thus obtain that f (λ1 , . . . , λn ) is a local norm for Ki /Q for all places of S. This is also the case for the real place because f (λ1 , . . . , λn ) > 0 in the case that there is a totally imaginary Ki . 2.2.3. Third step / S ∪ { } and we know Now, f (λ1 , . . . , λn ) = su is a unit in Zp for every Qp and p ∈ by [26, Prop. V.3.11] that this implies that f (λ1 , . . . , λn ) is a local norm for Ki /Q for all p∈ / S ∪ { }. By the global reciprocity law and the fact that Ki /Q is unramified outside S we see that f (λ1 , . . . , λn ) is also a local norm for Ki /Q at the place (see [26, Prop. V.12.9]). The conclusion is that f (λ1 , . . . , λn ) is a local norm for Ki /Q at every place of Q and then by the Hasse norm principle [26, Th. V.4.5], one gets that there exists (x1 , . . . , xr ) ∈ Qd1 × · · · × Qdr such that 0 = f (λ1 , . . . , λn ) = NK1 /Q (x1 ) = · · · = NKr /Q (xr ). 2.2.4. Fourth step × Bycontinuity,  there exists ε1 > 0 such that for all q1 ∈ Q such that |q1 − λ1 |∞ < ε1 , 1  ε then  q1 − λ11  < 2 max1in |λi |∞ . Writing m = [L, deg(f )], by weak approximation ∞

 in Q, since λ1 > 0 one can find ρ ∈ Q such that |ρ − 1|p < min 2ε , ε2 min{1, |λi |p }     for all p ∈ S and |ρm/ deg(f ) − λ1 |∞ < min 2ε min 1, λ1 − 2ε , ε1 where we can assume that ε < 2λ1 . In particular, this implies that |ρ|p = 1 for all p ∈ S. We now make the following change of variables, λi = ρm/ deg(f ) λi ,

i ∈ {1, . . . , n},

xi = ρm/di xi ,

i ∈ {1, . . . , r},

so that for all finite place p ∈ S we have |λi − λi |p = |λi |p |ρm/ deg(f ) − 1|p  |λi |p |ρ − 1|p <

ε 2

and therefore |λi − ti,p |p < ε for all i ∈ {1, . . . , n}. Moreover, we have 0 = f (λ1 , . . . , λn ) = NK1 /Q (x1 ) = · · · = NKr /Q (xr ).

(2.3)

As for the real place, we have |λ1 −1|∞ < ε and |λi −λi /λ1 |∞ < ε/2. The treatment of the archimedian place is now concluded similarly as in [20], by alluding to 0 < λi /λ1 < C −1 and by taking C big enough, namely C > 2ε . 2.2.5. Fifth step To conclude the proof, it remains to find (x1 , . . . , xr ) ∈ Qd1 × · · · × Qdr v-adically close to (x1,v , . . . , xr,v ) for all v ∈ S and such that NKi /Q (xi ) = NKi /Q (xi ) for all

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11

i ∈ {1, . . . , r}. By (2.2) and the choice of λ = (λ1 , . . . , λn ) above, one can write ε f (λ1 , . . . , λn ) = f (t1,p , . . . , tn,p )βp with βp ∈ Qp satisfying |βp − 1|p < 2M |L|p for all × finite place p ∈ S. In particular, |βp |p = 1 and βp ∈ Zp for all finite place p ∈ S. Now, Hensel’s lemma implies that there exists αp ∈ Zp such that βp = αpL and    βp − 1   < ε .  |αp − 1|p =  L p 2M L Of course, there exists α∞ such that β∞ = α∞ and |α∞ − 1|∞ < ε/(2M ) since one can  always ensure that f (1, 0, . . . , 0) and f (λ1 , . . . , λn ) have the same sign. Now alluding to the fact that (tv , x1,v , . . . , xr,v ) ∈ X(Qv ) and to (2.3), we obtain that for all v ∈ S

0 = f (λ1 , . . . , λn ) = NK1 /Q (αvL/d1 x1,v ) = · · · = NKr /Q (αvL/dr xr,v ).   L/d In other words, for every i ∈ {1, . . . , r}, we have NKi /Q αv i xi,v = NKi /Q (xi ) for all v ∈ S. Thanks to the fact that weak approximation holds for the norm tori NKi /Q (z) = 1, L/d one gets the existence of xi ∈ Qdi such that |xj,i − αv i xj,i,v |v < ε/2 for all v ∈ S and j ∈ {1, . . . , di } and NKi /Q (xi ) = NKi /Q (xi ). Therefore, we have 0 = f (λ1 , . . . , λn ) = NK1 /Q (x1 ) = · · · = NKr /Q (xr ) along with |xj,i − xj,i,v |v  |xj,i − αvL/di xj,i,v |v + |xj,i,v |v |αv − 1|v < ε, thus concluding the proof of Theorem 1.1.

2

3. The proof of Theorem 1.3 3.1. First steps and auxiliary estimates The proof of Theorem 1.3 starts by using the following exponential sums for real α, S(α) :=

 x∈Zn ∩P B



e (αf (x)), W (α) := 1 2

e(αp), (3.1)

min{f0 (B)}P d p2 max{f0 (B)}P d

where we used that, in the setting of Theorem 1.3, the following quantities are positive min{f0 (B )} = min{f0 (x) : x ∈ B } and max{f0 (B )} := max{f0 (x) : x ∈ B }. 1 The fact that 0 e(α{f (x) − p})dα is 1 when f (x) = p and is otherwise 0, shows that for all P f,B 1, we have the equality

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

12

1 πf (P B ) =

S(α)W (α)dα.

(3.2)

0

This identity has the useful feature that it completely separates the problem of evaluating πf into two problems, one regarding the evaluation of the sum S (that is only related to the values of the polynomial f ) and one regarding the evaluation of the sum W (that is only related to the distribution of primes). Birch [2] has a similar identity, save for the factor W (α). The main idea is that the presence of this extra factor can be turned to our advantage, as it attains small values for certain α for which |S(α)| is large. Let us comment that we could have defined W in an alternative way by replacing the range for the primes p by the condition min{f0 (B )}P d  p  max{f0 (B )}P d , however, our choice will make more transparent the proof of Lemma 3.19. Before proceeding let us recall here the estimates from the work of Birch [2] that we shall need later. First, following [2, pg. 251, Eq. (5)], we let for θ ∈ (0, 1] and a ∈ Z ∩[0, q) with gcd(a, q) = 1,

Ma,q (θ) := α ∈ (0, 1] : 2|qα − a|  P −d+(d−1)θ (3.3) and M (θ) =





Ma,q (θ).

(3.4)

1qP (d−1)θ a∈Z∩[0,q) gcd(a,q)=1

Birch then gives the following upper bound for the volume of M (θ). Lemma 3.1 (Birch [2, Lem. 4.2]). M (θ) has volume at most P −d+2(d−1)θ . Next, we choose any positive δ, θ0 satisfying 1 > δ + 6dθ0 and

n − σf − (d − 1) > δθ0−1 . 2d−1

(3.5)

As in [2, pg. 252, Eq. (13)–(14)] it is easy to see that there exists T ∈ N and positive real numbers θ1 , . . . , θT with the properties ⎧ T P δ, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ θT > θT −1 > . . . > θ1 > θ0 > 0, d = 2(d − 1)θT ⎪ ⎪ ⎪ ⎪ ⎪1 ⎪ ⎩ δ > 2(d − 1)(θ t+1 − θt ) for 0  t  T − 1. 2

(3.6)

We next recall [2, Lem. 4.3]. Note that it was proved for homogeneous f , but, as noted by Schmidt [31, §9] a similar argument works for inhomogeneous f , because the Weyl differencing process is not affected by lower order terms.

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

13

Lemma 3.2 (Birch [2, Lem. 4.3]). Let 0 < θ  1 and ε > 0. Then if α is not in M (θ) modulo 1, |S(α)| P

n−θ

 n−σ  f 2d−1

+ε

.

Following the notation in [2, pg. 253] and for θ, a, q as above we also let

 Ma,q (θ) := α ∈ (0, 1] : |qα − a|  qP −d+(d−1)θ

(3.7)

and 



M  (θ) =

 Ma,q (θ).

(3.8)

1qP (d−1)θ a∈Z∩[0,q) gcd(a,q)=1

With θ0 as in (3.5), we let η := (d − 1)θ0

(3.9)

and for a ∈ Z, q ∈ N we define Sa,q :=





e

x∈(Z/qZ)n

af (x) q

.

(3.10)

Finally, for any γ ∈ R and any measurable C ⊂ [−1, 1]n we define  I(C ; γ) :=

e (γf0 (x)) dx.

(3.11)

x∈C

The following result, due to Birch, gives an upper bound for the quantity I(C ; γ). Lemma 3.3 (Birch [2, Lem. 5.2]). Let C be a box contained in [−1, 1]n with sidelength at most σ < 1. Then   n−σ − d−1 f +ε n d 2 (d−1) . I (C , γ) σ min 1, (σ |γ|) The next result was proved by Birch with f instead of f0 in the definition of I(C ; γ), however the following result holds in light of the remarks concerning the function μ(∞, B ) appearing in Schmidt’s work [31, §9]. Lemma 3.4 (Birch [2, Lem. 5.1]). Assume that we are given coprime integers q ∈ N  (θ0 ) with the notations (3.5) and (3.7). Denoting and a ∈ Z ∩ [0, q) and let α ∈ Ma,q a β := α − q , we have

14

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

S(α) = q −n P n Sa,q I(B ; P d β) + O(P n−1+2η ). Let us now turn to the quantity Sa,q defined in (3.10). The next two lemmas will be used to prove Proposition 3.7. Lemma 3.5 (Birch [2, Lem. 5.4]). For every ε > 0 and for a ∈ Z, q ∈ N with gcd(a, q) = 1 we have Sa,q q

n−

n−σf +ε 2d−1 (d−1)

.

Lemma 3.6 (Browning–Heath-Brown [4, Lem. 25] ). We have Sa,pk k p(k−1)n+σf for all k  2. Note that, as explained in [4, Eq. (6.1)], the quantity σ in [4, Lem. 25] coincides with −1 + σf , with σf as in the present work. The next result is of key importance in the proof of Theorem 1.3. It is what allows to save variables compared to the Birch setting. Proposition 3.7. Let f ∈ Z[x1 , . . . , xn ] be an irreducible polynomial and define 

Tf (q) := q −n

|Sa,q |, q ∈ N.

a∈(Z/qZ)∗

(1) If n − σf  max{5, (deg(f ) − 1)2deg(f )−1 + 2} then the abscissa of convergence of the Dirichlet series of Tf is strictly negative. (2) If n − σf  max{4, (deg(f ) − 1)2deg(f )−1 + 1} then there exists a constant C  =   C  (f ) > 0 such that qx Tf (q) (log x)C .  Proof. Part (1). It suffices to show that q q λ1 Tf (q) < ∞ holds for some λ1 > 0. Owing to [2, §7], the function Tf is multiplicative, hence the series over q converges absolutely if the analogous Euler product converges absolutely, i.e. 

pkλ1 Tf (pk ) =

p prime k∈N



pk(λ1 −n)



|Sa,pk | < ∞.

(3.12)

a∈(Z/pk Z)∗

p prime k∈N

By Lemma 3.5 the terms with k > 2d−1 (d − 1) contribute





p k1+2d−1 (d−1)

p

 k 1−

n−σf +ε+λ1 2d−1 (d−1)



.

(3.13)

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

15

By the assumption n − σf  (d − 1)2d−1 + 2 2 n − σf 1+ d−1 (d − 1)2 (d − 1)2d−1 we have 

p

1−

n−σf +ε+λ1 2d−1 (d−1)





p

−

2 +ε+λ1 2d−1 (d−1)



,

thus taking ε, λ1 sufficiently small we can ensure that this is at most p

−

1 2d−1 (d−1)

2

−

1 2d−1 (d−1)

,

which is of the form 1 − δ for some 0 < δ < 1. Note that if δ ∈ (0, 1) then for all z ∈ R with 0  z  1 − δ and all k0 ∈ N we have 

zk =

kk0

z k0 z k0  . 1−z δ

Therefore the sum in (3.13) is d



p

 (1+2d−1 (d−1)) 1−

n−σf +ε+λ1 2d−1 (d−1)

p





p−2+(ε+λ1 )(1+2

d−1

(d−1))

p







p−3/2 < ∞,

p

where we have taken ε, λ1 sufficiently small to ensure (ε + λ1 )(1 + 2d−1 (d − 1))  1/2. Next, we study the contribution towards (3.12) of any k ∈ [2, 2d−1 (d − 1)]. By the result [4, Lem. 25] we infer that the said contribution is



pk(λ1 −n)+k+(k−1)n+σf =



p

p(1+λ1 )k−n+σf 

p



p(1+λ1 )(2

d−1

(d−1))−n+σf

.

p

The assumption n −σf  (d −1)2d−1 +2 shows that the exponent is  λ1 (2d−1 (d −1)) −2 and for small λ1 the sum converges. To conclude the proof of (3.12) it only remains to bound the contribution of terms with k = 1. As noted in [6, §5], one can prove Tf (p) = p−n



|Sa,p | p1−

n−σf 2

(3.14)

a∈(Z/pZ)∗

by Deligne’s estimate and induction on σf . Taking small λ1 < 1/4 and using the assumption n − σf  5 shows that the terms with k = 1 in (3.12) form a convergent series. This completes our proof.

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

16

Part (2). If k ∈ [2, 2d−1 (d − 1)] then Lemma 3.6 and n − σf  1 + 2d−1 (d − 1) imply that Tf (pk ) p−1 . Furthermore, using Lemma 3.5 and n − σf  1 + 2d−1 (d − 1) we have d −1 that if k  1 + 2d−1 (d − 1) then Tf (pk ) p−1−2 (d−1) . Finally, n − σf  4, thus (3.14)  ensures that Tf (p) p−1 . Putting everything together yields k1 Tf (pk )  C  p−1 for some C  = C  (f ) > 0 and the observation ⎞ ⎛    ⎝1 + Tf (q)  Tf (pk )⎠ qx

px

k1

concludes the proof. 2 3.2. The minor arcs For θ ∈ (0, 1] and a ∈ Z ∩ [0, q) with gcd(a, q) = 1 we use the sets M (θ) and Ma,q (θ) defined by (3.4) and (3.3). Next, we choose any positive δ, θ0 satisfying (3.5). Lemma 3.8. For any 0 < θ  1 we have   ⎛ ⎞1/2       ⎜   ⎟ S(α)W (α)dα ⎝ |S(α)|2 dα⎠ P d/2 (log P )−1/2 .     α∈/ M (θ)  α∈ / M (θ) Proof. By Schwarz’s inequality the integral on the left side is bounded by ⎛

⎞1/2 ⎛ ⎞1/2 1 ⎟ |S(α)|2 dα⎠ ⎝ |W (α)|2 dα⎠ .



⎜ ⎝

α∈ / M (θ)

0

The proof is concluded by noting that 1



|W (α)|2 dα = 1 2

0

min{f0

(B)}P d p2

1 P d / log P. max{f0

2

(B)}P d

Lemma 3.9. Keep the assumptions of Theorem 1.3 and (3.5). Then we have, ⎛



⎜ ⎝

⎞1/2 ⎟ |S(α)|2 dα⎠

= O(P n−d/2−δ/2 ).

α∈ / M (θ0 )

Proof. Using the entities (θi )Ti=0 , given in (3.6), we have for sufficiently small ε > 0,  |S(α)|2 dα P α∈ / M (θT )

  n−σ   2 n− d−1f θT +ε 2

 P 2n−d−δ ,

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

17

due to Lemma 3.2, the third equation of (3.6) and (1.4). For t < T and ε > 0 we get  |S(α)| dα P 2

  n−σ   −d+2(d−1)θt+1 +2 n− d−1f θt +ε 2

M (θt+1 )M (θt )

by Lemmas 3.1 and 3.2. The proof can now be completed easily by using the last equation of (3.6), (3.5) and T P δ , as in the last stage of the proof of [2, Lem. 4.4]. 2 Lemma 3.10. Keep the assumptions of Theorem 1.3 and (3.5). Then we have,          S(α)W (α)dα = O(P n−δ/2 ).      α∈/ M (θ0 ) Proof. The proof follows immediately by tying together Lemmas 3.8 and 3.9. 2  Recall the definition of M  (θ0 ) and Ma,q (θ0 ) given in (3.8) and (3.7). The next lemma is analogous to [2, Lem. 4.5].

Lemma 3.11. Keep the assumptions of Theorem 1.3 and (3.5). Then we have πf (P B ) =





 qP (d−1)θ0

S(α)W (α)dα + O(P n−δ/2 ),

a∈Z∩[0,q) M  (θ ) 0 gcd(a,q)=1 a,q

where C  is as in Proposition 3.7. Before proceeding we note that one can take an arbitrarily small positive value for θ0 in Lemma 3.11 because the system of inequalities (3.5) can be solved for any θ0 > 0 small enough. This will come at the cost of a worse error term in Lemma 3.11, however, it will still exhibit a power saving and it will thus be acceptable for the purpose of verifying Theorem 1.3. 3.3. The intermediate range Under the assumptions of Theorem 1.3 and (3.5) we can use Lemmas 3.1, 3.4 and the trivial bound W (α) P d to evaluate the quantity S(α) in Lemma 3.11. This yields πf (P B ) − Pn





qP (d−1)θ0 a∈Z∩[0,q) gcd(a,q)=1

Sa,q qn



|γ|P η

I(B ; γ)W ( aq + Pd

γ )dγ Pd

(log P )−A , (3.15)

valid for all A > 0, where η, Sa,q and I(B ; γ) are defined respectively in (3.9), (3.10) and (3.11).

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K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

For A, q ∈ N and a ∈ Z ∩ [0, q) with gcd(a, q) = 1 we let Ma,q (A) := {α ∈ R (mod 1) : |α − a/q|  P −d (log P )A },   M(A) := Ma,q (A)

(3.16) (3.17)

1q(log P )A a∈Z∩[0,q) gcd(a,q)=1

and we observe that M(A) ⊂ M  (θ0 ) for all P  1. We denote the difference by t(A) := M  (θ0 ) \ M(A).

(3.18)

The set t(A) is therefore to be thought of as lying ‘between’ the major arcs M  (θ0 ) and the minor arcs [0, 1) \ M  (θ0 ). We shall see in §3.4 that M(A) gives rise to the main term in Theorem 1.3. Next, we observe that Lemma 3.3 and our assumption n − σf  1 + 2d−1 (d − 1) yield 

−

|I(B ; γ)|dγ Q

1 2d (d−1)

,

(Q  1),

(3.19)

|γ|Q

in particular showing that

 R

|I(B ; γ)|dγ converges under assumption (1.4).

Lemma 3.12. If (1.4) holds then 

q −n

(log P )A
 a∈Z∩[0,q) gcd(a,q)=1

 |Sa,q | |γ|P η

|I(B ; γ)|

|W (a/q + γP −d )| dγ Pd



(log P )−A/2+3+C ,

(3.20)

where C  is as in Proposition 3.7. Proof. If α is not in the union of the sets {α (mod 1) : |α − a/q|  P −d+(d−1)θ0 } taken over all q ∈ N ∩ [1, (log P )A ] and a ∈ Z ∩ [0, q) with gcd(a, q) = 1, then by Dirichlet’s approximation theorem there are coprime integers 1  a  q  with q   P d−(d−1)θ0 and |α − a /q  |  P −d+(d−1)θ0 /q  . Thus we must have q  > (log P )A . Alluding to Vaughan’s estimate [12, §25] and using partial summation we obtain |W (α)| (P d q  −1/2 + P 4d/5 + (P d q  )1/2 )(log P )3  (P d (log P )−A/2 + P 4d/5 + P d−η/2 )(log P )3 , which is P d (log P )−A/2+3 . For each a and q as in (3.20) we get by (3.19) that  |I(B ; γ)| |γ|P η

|W (a/q + γP −d )| dγ (log P )−A/2+3 , Pd

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

19

hence by the second part of Proposition 3.7 we see that the sum over q in the lemma is 





 |Sa,q | (log P )−A/2+3  (log P )−A/2+3+C , qn

(log P )A
which conlcudes the proof. 2 Lemma 3.13. Assume (1.4). Then we have  q(log P )A







q −n

|Sa,q |

a∈Z∩[0,q) gcd(a,q)=1

(log log P )C

|I(B ; γ)|

|W (a/q + γP −d )| dγ Pd

(log P )A <|γ|P η



.

A

(log P ) 2d (d−1) Proof. The proof follows immediately by combining the bound W (α) P d , the inequality (3.19) for Q = (log P )A and the second part of Proposition 3.7. 2 Tying Lemmas 3.12 and 3.13 proves the following lemma. Lemma 3.14. Keep the assumptions of Theorem 1.3. Then there exists a strictly positive constant λ = λ(f ) such that for every fixed sufficiently large A > 0 we have  S(α)W (α)dα

Pn . (log P )Aλ

α∈t(A)

3.4. The major arcs Bringing together (3.15), (3.18), and Lemma 3.14 we see that under the assumptions of Theorem 1.3 there exists λ > 0 such that for all large A > 0 we have πf (P B ) − Pn

 q(log P )A

(log P )−Aλ .

q

−n

 a∈Z∩[0,q) gcd(a,q)=1

 I(B ; γ)

Sa,q

W (a/q + γP −d ) dγ Pd

|γ|(log P )A

(3.21)

Using Siegel–Walfisz’s theorem as in [12, pg. 147] we can show that there exists a constant c = c(A) > 0 such that if |β|  P −d (log P )A , q  (log P )A , a coprime to q and x is in the interval [P d/2 , P 2d ], then

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

20

 mx

μ(q) Λ(m)e(m(a/q + β)) − ϕ(q)

x



 e(βt)dt (1 + |β|x)x exp −c log P , (3.22)

2

where μ, ϕ and Λ denote the Möbius, Euler and von Mangoldt functions. We now see that  px

μ(q) (log p)e(p(a/q + β)) − ϕ(q)

x



 e(βt)dt (1 + |β|x)x exp −c log P

2

due to the estimate



Λ(m) x1/2 . Partial summation shows that W (a/q + β)

mx m=p

equals μ(q) ϕ(q)

  2 max{f

 12 min{f0 (B)}P d e(βt)dt e(βt)dt − 2 1 log( 12 max{f0 (B )}P d ) log( 2 min{f0 (B )}P d ) 0 (B)}P

d

2

d 2 max{f 0 (B)}P

− min{f0 (B)}P d

1 2

 u

 1  e(βt)dt du log u



2

 √  up to an error of size (1 + |β|P d )P d exp −c log P . Partial integration now yields ⎛ μ(q) ⎜ W (a/q + γP −d ) = ⎝ ϕ(q)

d 2 max{f 0 (B)}P

1 2

min{f0

⎞ −d

e(γP t)dt ⎟ ⎠+O log t

(B)}P d



(1 + |γ|)P d  √  exp c log P

 . (3.23)

The error term makes the following contribution towards (3.21), 



exp −c log P

 q(log

q

−n

P )A





|Sa,q |

a∈Z∩[0,q) gcd(a,q)=1

|I(B ; γ)|(1 + (log P )A )dγ

|γ|(log P )A

 √  and, by the second part of Proposition 3.7 this is exp −c log P (log P )A+1 , which   √ is obviously exp −c/2 log P . Hence, letting ΞA (P ) :=



μ(q) ϕ(q)q n

q(log P )A



Sa,q

a∈Z∩[0,q) gcd(a,q)=1

and 



d 2 max{f 0 (B)}P

I(B ; γ)

ΨA (P ) := |γ|(log

P )A

1 2

min{f0 (B)}P d

 e(−γP −d t) dt dγ, log t

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

21

we obtain the following result via (3.21). Lemma 3.15. Under the assumptions of Theorem 1.3 there exists λ = λ(f ) > 0 such that for every A > 0 we have πf (P B ) = ΞA (P )ΨA (P )P n−d + O(P n (log P )−Aλ ) for all sufficiently large P . 3.5. The non-archimedean densities If n − σf  3 then (3.14) along with the multiplicativity of Tf [2, §7] gives  |μ(q)| ϕ(q)q n q>x



 |μ(q)| (n−σf ) q 1− 2 +ε . ϕ(q) q>x

|Sa,q | 

a∈(Z/qZ)∗

Hence, for q ∈ N, the estimate q/ϕ(q) log log(4q) that can be found for example in [36, Th. 5.6] implies  |μ(q)| ϕ(q)q n q>x



|Sa,q | x−1/2+ε .

a∈(Z/qZ)∗

Therefore, we have ΞA (P ) =

∞  μ(q) n ϕ(q)q q=1



Sa,q + O((log P )−A/4 ).

a∈(Z/qZ)∗

The multiplicativity of the last sum over a shows that the above sum over q is where βp := 1 −

1 (p − 1)pn



 p

βp ,

Sa,p .

a∈(Z/pZ)∗

Finally, the following lemma is obtained by observing that 

Sa,p = −pn + p#{x ∈ Fpn : f (x) = 0}.

(3.24)

a∈(Z/pZ)∗

Lemma 3.16. If n − σf  3 then ΞA (P ) =

 p



#{x ∈ Fpn : f (x) = 0} 1− pn



1 1− p

−1 

+ O((log P )−A/4 ).

Combining (3.14) and (3.24) yields p−n #{x ∈ Fpn : f (x) = 0} = 1/p + O(p−(n−σf )/2 ), thus verifying the following lemma. Lemma 3.17. If n − σf  3 then the product in Theorem 1.3 converges absolutely.

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

22

3.6. The archimedean densities Letting for P d >

1 2

min{f0 (B )}, 



2 max{f  0 (B)}

I(B ; γ)

Ψ(P ) :=

1 2

γ∈R

 e(−γμ) dμ dγ, log(μP d )

min{f0 (B)}

we see by (3.19) and our assumption (1.4) that there exists λ2 = λ2 (f ) > 0 such that ΨA (P )P −d = Ψ(P ) + OA ((log P )−λ2 A ).

(3.25)

Now we observe that for all reals z, μ with z > μ > 0 and z ∈ / {1/μ, 1} we have ∞ 1 1 1 1  (−1)k = (log μ)k , =   μ k log(μz) log z 1 + log log z (log z) log z

(3.26)

k=0

therefore, letting for k ∈ Z0 , 

 I(B ; γ)

J(k) :=

k

e(−γμ)(log μ) dμ dγ, 1 2

γ∈R



2 max{f  0 (B)}

(3.27)

min{f0 (B)}

we infer that for all sufficiently large P we have ∞

Ψ(P ) =

 (−1)k 1 J(k). log(P d ) (log(P d ))k

(3.28)

k=0

Let us furthermore introduce the following entity for all n ∈ N and k ∈ Z0 ,  Jn (k) :=



π2 γ 2 exp − 2 n



I(B ; γ)

k

e(−γμ)(log μ) dμ dγ. 1 2

γ∈R



2 max{f  0 (B)}

(3.29)

min{f0 (B)}

Lemma 3.18. Under the assumption (1.4) we have lim Jn (k) = J(k) for every k ∈ Z0 . n→+∞

Proof. We have J(k) − Jn (k) k



 H† (γ) :=

|γ|log n

H† (γ)dγ +

 |γ|>log n

H† (γ)dγ, where

 π2 γ 2 |I(B ; γ)|. 1 − exp − 2 n

We have I(B ; γ) 1 due to Lemma 3.3, hence,

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

 |γ|log n

By (3.19) we get depends on f . 2

23

  π 2 (log n)2 = o(1). H† (γ)dγ (log n) 1 − exp − n2

 |γ|>log n

H† (γ)dγ (log n)−λ1 = o(1) for some positive λ1 which

Lemma 3.19. Under the assumption (1.4) we have the following for every k ∈ Z0 ,  (log f0 (t))k dt.

lim Jn (k) =

n→+∞

t∈B

Proof. It is standard to see that the Fourier transform of the function ϕn : R → R defined through ϕn (x) := π −1/2 n exp(−n2 x2 ) satisfies ϕ $n (γ) = exp(−π 2 n−2 γ 2 ). Therefore, the  Fourier inverse formula yields ϕn (x) = R e(xγ)ϕ $n (γ)dγ. Using this for x = f0 (t) − y and rewriting (3.29) as 

 

2 max{f  0 (B)} k

(log μ) t∈B

1 2

min{f0 (B)}

we infer that Jn (k) =

 B



π2 γ 2 exp − 2 n





e((f (t) − μ)γ)dγ dμdt,

γ∈R

gn (t)dt, where 2 max{f  0 (B)}

(log μ)k ϕn (f (t) − μ)dμ.

gn (t) := 1 2

min{f0 (B)}

It is obvious from [18, Ex. I.2] that for any reals a < c < b and any continuous function h : [a, b] → R one has b h(μ)ϕn (c − μ)dμ = h(c).

lim

n→+∞ a

Recalling that f0 (B ) ⊂ (0, ∞) we infer that whenever t ∈ B then the following inequality holds, 12 min{f0 (B )} < f0 (t) < 2 max{f (B )}. This gives limn→+∞ gn (t) = (log f0 (t))k and a use of the dominated convergence theorem concludes the proof of the lemma. 2 Lemma 3.20. Under the assumption (1.4) we have Ψ(P ) = P −n Lif (P B ) for all sufficiently large P .  Proof. Combining Lemmas 3.18 and 3.19 we get J(k) = B (log f0 (t))k dt. Injecting this into (3.28) and interchanging the sum over k and the integral over t yields

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

24

  Ψ(P ) =

 ∞  1 (−1)k k (log f0 (t)) dt. log(P d ) (log(P d ))k

(3.30)

k=0

B

The proof is concluded by making the change of variables x = P t and alluding to (3.26). 2 Combining Lemma 3.20 with (3.25) provides us with the following result. Lemma 3.21. Under the assumptions of Theorem 1.3 there exists λ2 = λ2 (f ) > 0 such that for every A > 0 and every sufficiently large P we have ΨA (P ) = Lif (P B )P −(n−d) + OA (P d (log P )−Aλ2 ). Our final result offers an asymptotic expansion of Lif (P B ) in terms of (log P )−1 . Lemma 3.22. For f and B as in Theorem 1.3 and P large enough we have

vol(B ) P + Pn d log P n

Lif (P B ) =

∞  (−1)k−1 k=2

dk

⎞

⎛



⎝ (log f0 (t))k−1 dt⎠ B

1 . (log P )k

In particular, we have

Lif (P B ) =

vol(B ) P n + Of,B d log P



Pn (log P )2

.

Proof. The first equality follows by combining Lemmas 3.18 and 3.19 with (3.28) and (3.30). To prove the second, note that if log P > 2 then ∞  (−1)k−1 k=2

dk

⎛



⎞

⎝ (log f0 (t))k−1 dt⎠ B

∞ ∞   1 1 1 1 < , k k 2 k−2 (log P ) (log P ) (log P ) 2 k=2

thus concluding the proof. 2 3.7. The proof of Theorem 1.3 It follows by merging Lemmas 3.15, 3.16 and 3.21. 2

k=2

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

25

4. The proof of Theorem 1.5 4.1. First steps and auxiliary estimates Similarly as in §3.1 we may write 1 #{Sf ∩ P B } =

S(α)Q(α)dα, 0

where S(α) is defined in (3.1) and 

Q(α) :=

e(αm).

m square-free,m=0 min{f0 (B)}−1mP −d max{f0 (B)}+1

We shall later need certain estimates concerning exponential sums taking values over square-free integers that we record here. For α ∈ R and N ∈ R1 define 

f2 (α, N ) :=

μ(n)2 e(αn).

1nN

The following result is the very special case corresponding to the choices k = 2 and p = 3/2 in the work of Keil [27]. Lemma 4.1 (Keil [27, Th. 1.2]). We have 1 |f2 (α, N )|3/2 dα N 1/2 (log N )2 . 0

For p prime and , m non-negative integers such that m  , define g(p , pm ) by ⎧ ⎪ ⎪ ⎨0, −2 m p (1 − p )g(p , p ) = 1, ⎪ ⎪ ⎩1 − p −2 ,

if  m  2, if m < min{2, }, if = m  1.

We extend this definition by defining the following whenever d, q ∈ N are such that d | q, g(q, d) :=



g(pνp (q) , pνp (d) ).

p|q

We can now introduce the following entity for q ∈ N,

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

26

G(q) :=

q 

e(b/q)g(q, gcd(b, q)).

(4.1)

b=1

Brüdern, Granville, Perelli, Vaughan and Wooley studied Q(α) in [7]. Lemma 4.2. There exist absolute positive constants δ1 , δ2 such that for all q ∈ N with q  P δ1 , all a ∈ Z ∩ [1, q), d ∈ N, γ ∈ R with |γ|  P δ1 and all c1 < c2 ∈ R we have ⎛ 

e(m(a/q+γP −d )) =

m square-free,m=0 c1 mP −d c2

G(q) ⎜ ⎝ ζ(2)

d c 2P

⎞   ⎟ e(γP −d t)dt⎠+Oc1 ,c2 (1 + |γ|)P d−δ2 ,

c1 P d

where ζ denotes the Riemann zeta function and the implied constant depends at most on c1 and c2 . Proof. We will show that there exists an absolute δ > 0 such that if |β|  P −d+δ , q  P δ , a is coprime to q and x ∈ [P d/2 , P 2d ] then  1mx

⎛ x ⎞    G(q) ⎝ e(βt)dt⎠ + O (1 + |β|x)x1−δ , μ(m)2 e(m(a/q + β)) = ζ(2)

(4.2)

1

from which one can deduce the asymptotic stated in the lemma in the same way as we deduced (3.23) from (3.22). To prove (4.2) we first note that for all b and q ∈ N we have 

1=

1mx m≡b(mod q) m square-free





μ(d) =

1mx d2 |m m≡b(mod q)

and completing the sum over d gives let Z(x; q, a) :=





 √

μ(d)

1d x gcd(q,d2 )|b

x ζ(2) g(q, gcd(b, q))

μ(m)2 e(ma/q) =

1mx

q 

x gcd(q, d2 ) + O (1) qd2

(4.3)

√ + O( x). For gcd(a, q) = 1 we

e(ba/q)

b=1



μ(m)2

1mx m≡b(mod q)

and use (4.1) and (4.3) to get the following estimate with an absolute implied constant for q  x, Z(x; q, a) −

√ G(q)x q x. ζ(2)

We therefore obtain by partial summation that

(4.4)

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794



27

x μ(m) e(m(a/q + β)) = e(xβ)Z(x; q, a) − 2πiβ 2

1mx

e(uβ)Z(u; q, a)du 1

and by (4.4) this becomes ⎞ ⎛ x   √  G(q) ⎝ e(βt)dt⎠ + O (1 + |β|x)q x , ζ(2) 1

with an absolute implied constant. This proves (4.2) with δ = (2 + d)−1 . Indeed, if d 1 d 1 q  P δ then the equality P δ = P 2 ( 2 −δ) and the bound P 2  x yield q  x 2 −δ , i.e. √ q x  x1−δ . 2 Finally, the next result is shown in the proof of [7, Lem. 3.1]. Lemma 4.3 (Brüdern, Granville, Perelli, Vaughan and Wooley, [7, Lem. 3.1]). The function G is multiplicative, supported in cube-free integers and satisfies for all prime p the identity G(p) = G(p2 ) = −p−2 (1 − p−2 )−1 . 4.2. Continuation of the proof Recalling the meaning of M (θ) and Ma,q (θ) in (3.4) and (3.3), we allude to Hölder’s inequality and Lemma 4.1 to obtain  ⎛  ⎞1/3 ⎛ ⎞2/3     1    ⎜  ⎟ S(α)Q(α)dα  ⎝ |S(α)|3 dα⎠ ⎝ |Q(α)|3/2 dα⎠      α∈/ M (θ) 0 α∈ / M (θ) ⎛ ⎜ ⎝

⎞1/3



⎟ |S(α)|3 dα⎠

P d/3 (log P )4/3 .

α∈ / M (θ)

The proof of Lemma 3.9 can be adapted straightforwardly to show that if 1 > δ + 6dθ0 and

n − σf 2 − (d − 1) > δθ0−1 2d−1 3

then ⎛ ⎜ ⎝

 α∈ / M (θ)

⎞1/3 ⎟ |S(α)|3 dα⎠

P n− 3 − 9 . d

δ

(4.5)

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

28

Let η := (d − 1)θ0 . Under the assumptions of Theorem 1.5 and for θ0 as in (4.5), one obtains the following inequality that is in analogy with Lemma 3.11, 

#{Sf ∩ P B } =





  δ S(α)Q(α)dα + O P n− 10 .

qP η a∈Z∩[0,q)  M (θ0 ) gcd(a,q)=1 a,q

Similarly as in the proof of (3.15), one may now acquire some δ1 = δ1 (f ) > 0 such that  #{Sf ∩ P B } − Pn δ qP

 1

a∈Z∩[0,q) gcd(a,q)=1

Sa,q qn



I(B ; γ)Q( aq +

|γ|P η

γ )dγ Pd

Pd

P −δ1 . (4.6)

By Lemma 4.2 we see that for suitably small δ1 and all a as in (4.6) and q  P δ1 one has ⎛ Q(a/q + γP −d ) =

G(q) ⎜ ⎝ ζ(2)

(max{f0 (B)}+1)P d

⎞   ⎟ e(γP −d t)dt⎠ + O (1 + |γ|)P d−δ2 .

(min{f0 (B)}−1)P d

Therefore, as in the proof of Lemma 3.15, we may infer that there exists a positive constant δ3 = δ3 (f ) such that the quantity ζ(2)P −n #{Sf ∩ P B } equals 

 G(q) qn δ

qP

1

 a∈Z∩[0,q) gcd(a,q)=1

  Sa,q |γ|P δ1

I(B ; γ) Pd

(B)}+1)P d (max{f0

e(−γP −d t)dtdγ, (4.7)

(min{f0 (B)}−1)P d

up to an error term which is O(P −δ3 ). We shall now use Lemma 4.3 to show that the sum over q forms an absolutely convergent series. Bringing into play (3.14) and [4, Lem. 25] we obtain the bounds |G(p)Tf (p)| p−1−(n−σf )/2 and |G(p2 )Tf (p2 )| p−n+σf . Hence, assuming n − σf  2, these two estimates allow to modify easily the proof of Proposition 3.7, thereby showing that the abscissa of convergence of the Dirichlet series of |G(q)|Tf (q) is strictly negative. This provides δ4 = δ4 (f ) > 0 such that for all x  2,  one has q>x |G(q)|Tf (q) x−δ4 , hence the sum over q in (4.7) is Π +O(P −ηδ4 ), where Π is

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794 ∞  G(q) q=1

=

qn  p



29

Sa,q

a∈Z∩[0,q) gcd(a,q)=1



1 1 − p−2 (1 − p−2 )−1 n p

 a∈Z∩(0,p)

1 Sa,p + 2n p





Sa,p2

 .

2

a∈Z∩[0,p ) gcd(a,p)=1

One can easily see, for example, by using orthogonality of characters of Z/p2 Z to detect the condition f (x) = 0, that    1 # x ∈ (Z/p2 Z)n : f (x) = 0 = p2(n−1) 1 + n p



Sa,p +

a∈Z∩(0,p)

1 p2n



 Sa,p2 ,

a∈Z∩[0,p2 ),pa

from which we can show that Π /ζ(2) is  p



  # x ∈ (Z/p2 Z)n : f (x) = 0 1− . p2n

This is in agreement with the infinite product in Theorem 1.5. To deal with the integral in (4.7) we observe that the transformation t = P d μ gives

P

−d

(max{f0 (B)}+1)P d

e(−γP

−d

max{f0 (B)}+1

e(−γμ)dμ min{1, |γ|−1 },

t)dt =

(4.8)

min{f0 (B)}−1

(min{f0 (B)}−1)P d

hence Lemma 3.3 shows that the integral in (4.7) converges absolutely and equals 



max{f0 (B)}+1

I(B ; γ)

 e(−γμ)dμ dγ + O(P −δ5 )

(4.9)

min{f0 (B)}−1

γ∈R

for some δ5 = δ5 (f ) > 0. One can combine the bound (4.8) with Lemma 3.3 to show that the integral over γ in (4.9) equals vol(B ) using arguments that are entirely analogous with the case k = 0 in Lemmas 3.18 and 3.19. Thereby alluding to the well-known estimate #{Zn ∩ P B } = vol(B )P n + OB (P n−1 ) allows us to conclude the proof of Theorem 1.5. Declaration of competing interest There is nothing to declare.

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

30

Appendix A. The Bateman–Horn heuristics in many variables In this section we extend the Bateman–Horn heuristics from the setting of univariate polynomials to that of polynomials with arbitrarily many variables; we do so because we were unable to find a reference for this extension in the literature. In 1958, Schinzel [30] formulated the following conjecture concerning prime values of univariate polynomials. Conjecture A.1 (Schinzel’s hypothesis H, [30]). Let f1 , . . . , fr ∈ Z[x] be univariate irrer ducible polynomials with positive leading coefficient. If i=1 fi has no repeated polynomial factors and, for every prime p, there exists xp ∈ Z such that p  f1 (xp ) · · · fr (xp ), then there exist infinitely many integers m such that f1 (m), . . . , fr (m) are all primes. This conjecture was later refined by Bateman and Horn [1] who, based on the Cramér model and the heuristics behind the Hardy–Littlewood conjecture (see [34, pg. 6-8]), gave a quantitative version of Schinzel’s conjecture. Conjecture A.2 (Bateman–Horn’s conjecture, [1]). Keep the assumptions of Conjecture A.1. Then the number of integers m ∈ [1, P ] such that every f1 (m), . . . , fr (m) is prime is asymptotically equivalent to the following quantity as P → +∞, 



p prime

(1 − p−1 #{x ∈ Fp : f1 (x) · · · fr (x) = 0}) (1 − 1/p)r



1 deg(f1 ) · · · deg(fr )

P

dx . (log x)r

2

The convergence of the infinite product is established in [1] using the prime ideal theorem. These two conjectures lie very deep and imply a number of notoriously difficult conjectures as immediate corollaries (the twin primes conjecture among others; see [30] for a non exhaustive list of implications). There are applications to the arithmetic of algebraic varieties, see [9], [35] or [20], where Schinzel’s hypothesis is assumed in order to prove that the Hasse principle and weak approximation holds. Recall that for a polynomial f we denote by f0 the top degree part of f . Let us now record the multivariable version of the Bateman–Horn conjecture. Conjecture A.3 (Extension of the Bateman–Horn conjecture). Assume that we are given r irreducible polynomials f1 , . . . , fr ∈ Z[x1 , . . . , xn ] such that i=1 fi has no repeated polynomial factors. Moreover, we assume that B ⊂ Rn is a non-empty box such that fi0 (B ) ⊂ (1, ∞) for all i ∈ {1, . . . , r}. Denote by πf1 ,...,fr (P B ) the cardinality of the set of integer vectors x ∈ Zn ∩ P B for which every f1 (x), . . . , fr (x) is a positive prime number. Then πf1 ,...,fr (P B ) is asymptotic to the following quantity as P → +∞, 

 p prime

(1 − p−n #{x ∈ Fpn : f1 (x) · · · fr (x) = 0}) (1 − 1/p)r



dx . i=1 log fi0 (x)

r PB

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

31

Remark A.4. Before providing the heuristics behind Conjecture A.3 let us note that one can prove that the product over p converges. Indeed, a version of the prime number theorem for schemes over Z that can be found in the work of Serre [32, Cor. 7.13] implies that 

 xn #{x ∈

Fpn

: f1 (x) · · · fr (x) = 0} = r

px

dt log t



  √ + O xn e−c log x

2

for some c = c(f1 , . . . , fr ) > 0. Now partial summation implies that 

p

−n

 #{x ∈

Fpn

: f1 (x) · · · fr (x) = 0} = r log log x + C1 + O

px

1 log x



for some constant C1 . Hence the following series converges, 

p

−n

#{x ∈

p

Fpn

r : f1 (x) · · · fr (x) = 0} − p

,

which in turn yields the convergence of the product over p appearing in the statement of the conjecture. We end this section by adapting the heuristics behind Conjecture A.2 to the multivariate case. Recall that the Cramér model asserts that a random positive integer m of size X has probability 1/ log X of being a prime. An analogous statement can be made if the extra condition that m lies in a primitive arithmetic progression modulo q for some positive integer q is added, in this case the probability is 1/(ϕ(q) log X) owing to Dirichlet’s theorem on primes in arithmetic progressions. This implies that for coprime a, q, the conditional probability that a positive integer m of size X is prime provided that m ≡ a (mod q) equals Prob[m ∼ X is a prime | m ≡ a (mod q)] ≈

q 1/(ϕ(q) log X) = . 1/q ϕ(q) log X

(A.1)

In the setting of Conjecture A.2 observe that for typical x ∈ Zn the integer fi (x) can be prime only if fi (x) is coprime to all small primes. Therefore, letting z = z(P ) be a  function that slowly tends to infinity with P and letting P := pz p, we see that πf1 ,...,fr (P B ) ≈ #{Zn ∩ P B }



Prob[xi ≡ ai (mod P ) for all 1  i  n] · Pa,P ,

a∈(Z/P Z) ∀i∈{1,...,r}, fi (a)∈(Z/P Z)× n

(A.2) where Pa,P denotes the joint probability defined through

K. Destagnol, E. Sofos / Bull. Sci. math. 156 (2019) 102794

32

Pa,P := Prob[mi ∼ P deg(fi ) is a prime for all 1  i  r | mi ≡ fi (a) (mod P )]. This is because the integer fi (x) is typically of size P deg(fi ) when x ∈ P B and the values fi (x) are thought to behave like a random integer mi lying in the arithmetic progression fi (a) (mod P ), provided that x ≡ a (mod P ). Note that for i = j the polynomials fi  and fj are coprime due to the assumption that i fi has no repeated factors, therefore it is reasonable to expect that for i = j the integer values fi (x) and fj (x) behave independently. This suggests that Pa,P =

r 

Prob[mi ∼ P deg(fi ) is a prime | mi ≡ fi (a) (mod P )]

i=1

r and by (A.1) one now gets Pa,P = P r ϕ(P )−r (log P )−r i=1 (deg(fi ))−1 . Substituting this into (A.2) and noting that Prob[xi ≡ ai (mod P )] = 1/P yields r 1 πf1 ,...,fr (P B )  P 1 r ≈ n n vol(B )P ϕ(P ) log P i=1 deg(fi ) P



1.

a∈(Z/P Z)n ∀i∈{1,...,r}, fi (a)∈(Z/P Z)×

The sum over a forms a multiplicative function of P that can be evaluated as   pn − #{x ∈ Fpn : f1 (x) · · · fr (x) = 0} . pz

Putting everything together shows that we expect πf1 ,...,fr (P B ) to be approximated by  vol(B )P n r r (log P ) i=1 deg(fi )

pz



p p−1

r 

pn − #{x ∈ Fpn : f1 (x) · · · fr (x) = 0} pn

 .

In view of Remark A.4 the product over p  z(P ) converges to the product in Conjecture A.3 as P → +∞. For x ∈ P B we have fi0 (x) P deg(fi0 ) and using deg(fi ) = deg(fi0 ) we get   1dx vol(B )P n dx PB  r 

= , r r deg(fi ) ) (log P )r i=1 deg(fi ) log fi0 (x) log(P i=1 i=1 PB

thereby concluding our explanation of the asymptotic in Conjecture A.3. References [1] P. Bateman, R.A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comput. 16 (1962) 363–367. [2] B.J. Birch, Forms in many variables, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 265 (1962) 245–263.

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