Zeroes of polynomials with prime inputs and Schmidt's h -invariant
aa r X i v : . [ m a t h . N T ] M a r ZEROES OF POLYNOMIALS WITH PRIME INPUTSAND SCHMIDT’S h -INVARIANT STANLEY YAO XIAO AND SHUNTARO YAMAGISHI
Abstract.
In this paper we show that a polynomial equation admits infinitely many prime-tuplesolutions assuming only that the equation satisfies suitable local conditions and the polynomialis sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related tothe h -invariant introduced by W. M. Schmidt. Our results prove a conjecture of B. Cook and ´A.Magyar [3] for hypersurfaces of degrees 2 and 3. Introduction
Solving systems of integral polynomial equations in integers is among the oldest and persistentlyinteresting problems in number theory. It is understood, especially in the context of the Hardy-Littlewood circle method, that systems tend to become easier to solve when the number of variablesinvolved increases. For instance, it is not known whether the equation x + 1 = p where x variesin the integers and p varies among the primes has infinitely many solutions, but the corresponding3-variable equation x + y = p was solved by Fermat using elementary means over three centuriesago. One can then ask whether it is possible to interpolate between these situations. That is,given a system of polynomial equations which is solvable in the integers, one can ask whether thesystem remains solvable when some of the variables are restricted to a thin subset of integers.One particular natural subset is the set of prime numbers. Indeed, many interesting problemsinvolving prime numbers may be phrased in such a manner. For example, the existence of infinitelymany solutions to the equation x − y = 2 with x, y restricted to primes is precisely the twin primeconjecture.In [3], B. Cook and ´A. Magyar broke new ground by applying the Hardy-Littlewood circle methodto show, in great generality, that systems of polynomial equations in many variables can be solvedwhen all of the inputs are prime numbers. The key hypothesis they require is that the so-calledBirch singular locus must be sufficiently small. For f = { f , . . . , f r d } ⊆ Q [ x , . . . , x n ] a systemof forms (homogeneous polynomials) of degree d , we define the Birch singular locus V ∗ f to be theaffine variety in A n C given by V ∗ f = n x ∈ C n : rank (cid:18) ∂f r ( x ) ∂x j (cid:19) ≤ r ≤ r d ≤ j ≤ n < r d o , and let the Birch rank to be B ( f ) = n − dim V ∗ f . The Birch rank is an important invariant that arosein [1]. In [16], W. M. Schmidt introduced a different invariant, now called Schmidt’s h -invariant,for systems of polynomials. B. Cook and ´A. Magyar conjectured in [3, pp. 736] that their maintheorem ought to hold assuming the largeness of the h -invariant instead of the Birch rank (see(2.3)).In this paper, we give a partial solution to the conjecture of B. Cook and ´A. Magyar. Weestablish the conjecture for hypersurfaces with an additional assumption. However, our assumptionis redundant for quadratic polynomials and cubic polynomials; therefore, we establish the conjecture Date : Revised on October 12, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Hardy-Littlewood circle method, diophantine equations, prime numbers. for quadratic and cubic polynomials. Given a form f ∈ Q [ x , . . . , x n ] of degree at least 2, we definethe h -invariant h ( f ) of f to be the least positive integer h such that f can be written identicallyas(1.1) f = U V + · · · + U h V h , where each U i and V i are forms in Q [ x , . . . , x n ] of degree at least 1 (1 ≤ i ≤ h ). We then definethe following quantity h ⋆ ( f ) = max( |{ U i : deg U i = 1 }| ) , where the maximum is over all representations of the shape (1.1). In other words, h ⋆ ( f ) is themaximum number of linear forms involved in the representation of f as a sum of h = h ( f ) productsof rational forms. Clearly, we have h ⋆ ( f ) ≤ h ( f ) . For a degree d polynomial b ( x ) ∈ Q [ x , . . . , x n ],we define h ( b ) = h ( f ) where f ( x ) is the degree d portion of b ( x ). We note that any polynomial b ( x ) of degree 2 or degree 3 satisfies h ( b ) = h ⋆ ( b ) . We define the following quantity M b ( N ) = X x ∈ [0 ,N ] n ∩ Z n δ b ( x ) , where δ b ( x ) = Q ≤ i ≤ n log p i if x i = p t i i , p i is prime , t i ∈ N (1 ≤ i ≤ n ) and b ( x ) = 0 , . Let Λ be the von Mangoldt function, where Λ( x ) is log p if x is a power of a prime p and 0otherwise. We use the notation e ( x ) to denote e πix . We define(1.2) T ( b ; α ) = X x ∈ [0 ,N ] n ∩ Z n Λ( x ) e ( α · b ( x )) , where Λ( x ) = Λ( x ) · · · Λ( x n )for x = ( x , . . . , x n ) ∈ ( Z ≥ ) n . By the orthogonality relation, we have(1.3) M b ( N ) = X x ∈ [0 ,N ] n ∩ Z n δ b ( x ) = Z T ( b ; α ) dα. We obtain the following theorem by estimating the integral in (1.3).
Theorem 1.1.
Let b ( x ) ∈ Z [ x , . . . , x n ] be a polynomial of degree d . Then there exists a positivenumber A d dependent only on d such that the following holds. If h ⋆ ( b ) > A d , then there exist c > and C b such that M b ( N ) = C b N n − d + O (cid:18) N n − d (log N ) c (cid:19) . In fact, we prove that C b > b ( x ) = 0 has a non-singular solution in Z × p ,the units of p -adic integers, for every prime p and the equation f ( x ) = 0, where f ( x ) is the degree d portion of b ( x ), has a non-singular real zero in the interior of B = [0 , n .The following result is an immediate consequence of Theorem 1.1, which replaces the assumptionof large Birch rank in [3, Theorem 1] with large h -invariant for quadratic and cubic polynomials. Corollary 1.2.
Let d = 2 or , and b ( x ) ∈ Z [ x , . . . , x n ] be a polynomial of degree d . Then thereexists a positive number A d dependent only on d such that the following holds. If h ( b ) > A d , thenthere exist c > and C b such that M b ( N ) = C b N n − d + O (cid:18) N n − d (log N ) c (cid:19) . We establish Theorem 1.2 in a similar manner to [3], but we shall make use of the fact that therepresentation (1.1) has enough linear terms. We also modify the method in [3] to better suit ourpurposes, so that it is in terms of the h -invariant instead of the Birch rank.Despite Theorem 1.1 and Corollary 1.2 being our primary goals in this paper, it is necessary forus to work over a system of polynomials at times. Indeed, our strategy is to decompose a polynomialinto a sum of elements in a suitable system of polynomials, and then use methods which apply tosystems to deduce results of a single polynomial.The organization of the rest of the paper is as follows. In Section 2, we prove some basicproperties of the h -invariant. A sufficiently large h ⋆ ( b ) allows us to massage our polynomial b ( x )into something amenable to the circle method, through a process called the ‘regularization’. Wecollect results related to the regularization process in Section 3. In Section 4, we obtain resultsfrom [16] based on Weyl differencing in terms of polynomials instead of forms as in [16]. We choseto present the details in Section 4 to make certain dependency of the constants explicit, becauseit plays an important role in our estimates. We then obtain the minor arc estimates in Section 5,and the major arc estimates in Section 6. Acknowledgments.
We would like to thank D. Schindler and the anonymous referees for manyhelpful comments. We would also like to thank the department of Pure Mathematics at Universityof Waterloo for their support as portions of this work were completed while both of the authorswere there as graduate students.2.
Properties of the h -invariant Let f = { f , . . . , f r d } ⊆ Q [ x , . . . , x n ] be a system of forms of degree d >
1. We generalize thedefinition of h -invariant for a single form, and define the h -invariant of f by(2.1) h ( f ) = min µ ∈ Q rd \{ } h ( µ f + · · · + µ r d f r d ) . Given an invertible linear transformation T ∈ GL n ( Q ), let f ◦ T = { f ◦ T, . . . , f r d ◦ T } . It followsfrom the definition of the h -invariant that h ( f ) = h ( f ◦ T ) . Let b = ( b , . . . , b r d ) ⊆ Q [ x , . . . , x n ] be a system of degree d polynomials. We let f r to be thedegree d portion of b r (1 ≤ r ≤ r d ), and define(2.2) h ( b ) = h ( { f r : 1 ≤ r ≤ r d } ) . It is known that large Birch rank implies large h -invariant, since we have h ( f ) ≥ − d B ( f )(2.3)by [16, Lemma 16.1, (10.3), (17.1)].We prove two basic lemmas regrading the properties of the h -invariant in this section. Let f ∈ Q [ x , . . . , x n ] be a form of degree d . For 1 ≤ i ≤ n , let f | x i =0 = f ( x , . . . , x i − , , x i +1 , . . . , x n ) ∈ Q [ x , . . . , x n ], which is either identically 0 or a form of degree d . Let h ( f ) = 0 if f is identically 0.We prove the following simple lemma. STANLEY YAO XIAO AND SHUNTARO YAMAGISHI
Lemma 2.1.
Let f ∈ Q [ x , . . . , x n ] be a form of degree d > . Then for any ≤ i ≤ n , we have h ( f ) − ≤ h ( f | x i =0 ) ≤ h ( f ) . Proof.
Without loss of generality, we consider the case i = 1. Let us write(2.4) f ( x , . . . , x n ) = x g ( x , . . . , x n ) + f (0 , x , . . . , x n ) . Clearly, g ( x ) is either identically 0 or a form of degree d −
1. Let h = h ( f ) and h ′ = h ( f | x =0 ).By the definition of h -invariant, we can find rational forms U j ′ , V j ′ (1 ≤ j ′ ≤ h ′ ) of positive degreethat satisfy f (0 , x , . . . , x n ) = U V + · · · + U h ′ V h ′ . Note if h ′ = 0, we assume the right hand side to be identically 0. By substituting the aboveequation into (2.4), we obtain f = x g + U V + · · · + U h ′ V h ′ . Because g ( x ) is either identically 0 or a form of degree d −
1, it follows that h ≤ h ′ . For the other inequality, let u j , v j (1 ≤ j ≤ h ) be rational forms of positive degree that satisfy(2.5) f = u v + · · · + u h v h . By substituting x = 0 into each form on both sides of the equation, it is clear that we obtain h ′ ≤ h . This completes the proof of the lemma. We add a remark that in the special case when f satisfies f = x v + u v + · · · + u h v h , in other words when we have u = x in (2.5), we easily obtain h ′ = h − (cid:3) The following is an immediate consequence of Lemma 2.1.
Lemma 2.2.
Let f = { f , . . . , f r d } ⊆ Q [ x , . . . , x n ] be a system of forms of degree d > . Suppose h ( f ) > . Then for any ≤ i ≤ n , we have h ( f ) − ≤ h ( f | x i =0 ) ≤ h ( f ) , where f | x i =0 = { f | x i =0 , . . . , f r d | x i =0 } . Let f ( x ) ∈ Q [ x , . . . , x n ] be a form, and let h = h ( f ) and 0 < M ≤ h . Suppose we have f = u V + · · · + u M V M + U M +1 V M +1 + · · · + U h V h , where each u i is a linear rational form (1 ≤ i ≤ M ), and each U i ′ and V j are rational forms ofpositive degree ( M + 1 ≤ i ′ ≤ h, ≤ j ≤ h ). It can be easily verified that the linear forms u , . . . , u M are linearly independent over Q . Then by considering the reduced row echelon form ofthe matrix formed by the coefficients of u , . . . , u M , and relabeling the variables if necessary, wemay suppose without loss of generality that(2.6) f = ( x + ℓ ) v + · · · + ( x M + ℓ M ) v M + u M +1 v M +1 + · · · + u h v h , where each ℓ i is a linear form in Q [ x M +1 , . . . , x n ] (1 ≤ i ≤ M ), and each u i ′ and v j are rationalforms of positive degree ( M + 1 ≤ i ′ ≤ h, ≤ j ≤ h ). We then define g M ∈ Q [ x , . . . , x n ] in thefollowing manner, f ( x , x , . . . , x n ) = g M ( x , . . . , x n ) + f ( − ℓ , . . . , − ℓ M , x M +1 , . . . , x n ) . (2.7)We note that there is no ambiguity for defining the polynomial f ( − ℓ , . . . , − ℓ M , x M +1 , . . . , x n ) ∈ Q [ x M +1 , . . . , x n ]obtained by substitution, because each ℓ i ∈ Q [ x M +1 , . . . , x n ] (1 ≤ i ≤ M ). It is also clear that(2.8) g ( − ℓ , . . . , − ℓ M , x M +1 , . . . , x n ) = 0 . Lemma 2.3.
Let ≤ M ≤ h . Suppose a degree d form f ( x ) ∈ Q [ x , . . . , x n ] satisfies (2.6). Define g M ( x ) as in (2.7). Then we have h ( g M ) ≥ M and h ( f ( − ℓ , − ℓ , . . . , − ℓ M , x M +1 , . . . , x n )) = h − M. Proof.
Since the linear forms ( x − ℓ ) , . . . , ( x M − ℓ M ) are linearly independent over Q , we can find A ∈ GL n ( Q ) such that x − ℓ ... x M − ℓ M x M +1 ... x n = A ◦ x ... x n . Let e f ( x ) = f ( A ◦ x ) . We then have e f ( A − ◦ x ) = f ( x ) , and also that h ( e f ) = h ( e f ◦ A − ) = h ( f ) = h . Because f ( x ) satisfies (2.6), it follows that e f ( x )satisfies e f = x V + · · · + x M V M + U M +1 V M +1 + · · · + U h V h , where each U i and V j are rational forms of positive degree ( M + 1 ≤ i ≤ h, ≤ j ≤ h ).Recall each ℓ i is a linear form in Q [ x M +1 , . . . , x n ] (1 ≤ i ≤ M ). Clearly, we have e f (0 , . . . , , x M +1 , . . . , x n ) = f ( A ◦ (0 , . . . , , x M +1 , . . . , x n ))= f ( − ℓ , − ℓ , . . . , − ℓ M , x M +1 , . . . , x n ) . Then we can deduce from Lemma 2.1 (see the remark at the end of the proof of Lemma 2.1) that h ( f ( − ℓ , − ℓ , . . . , − ℓ M , x M +1 , . . . , x n )) = h ( e f (0 , . . . , , x M +1 , . . . , x n )) = h − M. It then follows easily from the fact that h ( f ) = h , the definition of h -invariant, and (2.7), that h ( g M ) ≥ M, for otherwise we obtain a contradiction. (cid:3) Regularization lemmas
In this section, we collect results from [3] and [16] related to regular systems (see Definition 3.1)and the regularization process (Proposition 3.5), which played an important role in [3] to obtain theminor arc estimate. Throughout this section we use the following notation. Let d, n >
1, and let f be a system of forms in Q [ x , . . . , x n ] of degree less than or equal to d . We denote f = ( f ( d ) , . . . , f (1) ),where f ( i ) is the subsystem of all forms of degree i in f (1 ≤ i ≤ d ). We label the elements of f ( i ) by f ( i ) = { f ( i )1 , . . . , f ( i ) r i } , where r i = | f ( i ) | , the number of elements in f ( i ) .We shall call a system of polynomials regular if it has at most the expected number of integersolutions, which we define formally below. STANLEY YAO XIAO AND SHUNTARO YAMAGISHI
Definition 3.1.
Let d >
1. Let ψ = ( ψ ( d ) , . . . , ψ (1) ) be a system of polynomials in Q [ x , . . . , x n ],where ψ ( i ) is the subsystem of all polynomials of degree i in ψ (1 ≤ i ≤ d ). We denote V ψ , ( Z ) tobe the set of solutions in Z n of the equations ψ ( i ) j ( x ) = 0 (1 ≤ i ≤ d, ≤ j ≤ | ψ ( i ) | ) , which we denote by ψ ( x ) = . Let r i = | ψ ( i ) | (1 ≤ i ≤ d ), and let D ψ = P di =1 ir i . We say thesystem ψ is regular if | V ψ , ( Z ) ∩ [ − N, N ] n | ≪ N n − D ψ . Similarly as above we also define V ψ , ( R ) to be the set of solutions in R n of the equations ψ ( x ) = .The following is one of the main results of [16] which provides a sufficient condition for a systemof polynomials to be regular. Theorem 3.2 (Schmidt, [16]) . Let d > . Let ψ = ( ψ ( d ) , . . . , ψ (2) ) be a system of rationalpolynomials with notation as in Definition 3.1, and also let f ( i ) be the system of degree i portion ofthe polynomials ψ ( i ) (2 ≤ i ≤ d ) . We denote r i = | ψ ( i ) | = | f ( i ) | (2 ≤ i ≤ d ) , and R ψ = P di =2 r i . Ifwe have h ( f ( i ) ) ≥ d i ( i !) r i R ψ (2 ≤ i ≤ d ) , then the system ψ is regular. Let us denote(3.1) ρ d,i ( t ) = d i ( i !) t (2 ≤ i ≤ d )so that for each 2 ≤ i ≤ d , we have ρ d,i ( t ) is an increasing function, and ρ d,i ( R ψ ) ≥ d i ( i !) r i R ψ . Note Theorem 3.2 is regarding a system of polynomials that does not contain any linear poly-nomials. We prove Corollary 3.3 for systems that contain linear forms as well. Note the content ofthe following Corollary 3.3 is essentially [3, Corollary 3].
Corollary 3.3.
Let d > . Let ψ = ( ψ ( d ) , . . . , ψ (1) ) be a system of rational polynomials withnotation as in Definition 3.1. Suppose ψ (1) only contains linear forms and that they are linearlyindependent over Q . We also let f ( i ) be the system of degree i portion of the polynomials ψ ( i ) (1 ≤ i ≤ d ) . We denote r i = | ψ ( i ) | = | f ( i ) | (1 ≤ i ≤ d ) , and R ψ = P di =1 r i . For each ≤ i ≤ d , let ρ d,i ( · ) be as in (3.1). If we have h ( f ( i ) ) ≥ ρ d,i ( R ψ − r ) + r (2 ≤ i ≤ d ) , then the system ψ is regular.Proof. We have ψ (1) = f (1) = { f (1)1 , . . . , f (1) r } . Let f (1) i = a i x + · · · + a in x n (1 ≤ i ≤ r ) , and denote the coefficient matrix of these linear forms to be A = [ a ij ] ≤ i ≤ r , ≤ j ≤ n . Let e j be the j -th standard basis of R n (1 ≤ j ≤ n ). Since the linear forms f (1)1 , . . . , f (1) r are linearly independentover Q , we can find an invertible linear transformation T ∈ GL n ( Q ), where every entry of thematrix is in Z , such that ( f (1) i ◦ T − )( x ) = m n − i +1 x n − i +1 , where m n − i +1 ∈ Q \{ } (1 ≤ i ≤ r ).For simplicity, let us denote x ′ = ( x n − r +1 , . . . , x n ). Let Y = V f (1) , ( R ) = { x ∈ R n : f (1) ( x ) = } = { x ∈ R n : A ◦ x = } = Ker( A ) , which is a subspace of codimension r . Since T ( Y ) = Ker( A ◦ T − ), it follows from our choice of T ∈ GL n ( Q ) that T ( Y ) = R e + · · · + R e n − r . We also know there exit c ′ , C ′ > − c ′ N, c ′ N ] n ⊆ T ([ − N, N ] n ) ⊆ [ − C ′ N, C ′ N ] n . Define ψ ′ = ( ψ ′ ( d ) , . . . , ψ ′ (1) ) = ψ ◦ T − , and let f ′ ( i ) be the system of degree i portion of thepolynomials ψ ′ ( i ) (1 ≤ i ≤ d ). We then have f ′ ( i ) = f ( i ) ◦ T − . We can also verify that V ψ ′ , ( R ) = T ( V ψ , ( R )) . Therefore, we obtain T ( V ψ , ( R ) ∩ [ − N, N ] n ) ⊆ V ψ ′ , ( R ) ∩ [ − C ′ N, C ′ N ] n , and since every entry of the matrix T ∈ GL n ( Q ) is in Z , it follows that | V ψ , ( Z ) ∩ [ − N, N ] n | ≤ | V ψ ′ , ( Z ) ∩ [ − C ′ N, C ′ N ] n | . (3.2)Let ψ ′′ = ( ψ ′ ( d ) | x ′ = , . . . , ψ ′ (2) | x ′ = ). Since ψ ′ (1) = is equivalent to x ′ = , we have(3.3) | V ψ ′ , ( Z ) ∩ [ − C ′ N, C ′ N ] n | = | V ψ ′′ , ( Z ) ∩ [ − C ′ N, C ′ N ] n − r | . Since the degree i portion of ψ ′ ( i ) | x ′ = is f ′ ( i ) | x ′ = for each 2 ≤ i ≤ d , we have by Lemma 2.2that h ( f ′ ( i ) | x ′ = ) ≥ h ( f ′ ( i ) ) − r = h ( f ( i ) ) − r ≥ ρ d,i ( R ψ − r ) . Thus, it follows by Theorem 3.2 that(3.4) | V ψ ′′ , ( Z ) ∩ [ − C ′ N, C ′ N ] n − r | ≪ N ( n − r ) − P di =2 ir i . Therefore, we obtain from (3.2), (3.3), and (3.4) that | V ψ , ( Z ) ∩ [ − N, N ] n | ≪ N n − P di =1 ir i . (cid:3) Given g = { g , . . . , g r d } ⊆ Q [ x , . . . , x n ], a system of forms of degree d , and a partition ofvariables x = ( y , z ), we denote g to be the system obtained by removing all the forms of g thatdepend only on the z variables. Clearly, if we have the trivial partition x = ( y , z ), where z = ∅ ,then g = g . For a form g ( x ) over Q , we define h ( g ; z ) to be the smallest number h such that g ( x )can be expressed as g ( x ) = g ( y , z ) = h X i =1 u i v i + w ( z ) , where u i , v i are rational forms of positive degree (1 ≤ i ≤ h ), and w ( z ) is a rational form only inthe z variables. We also define h ( g ; z ) to be h ( g ; z ) = min λ ∈ Q rd \{ } h ( λ g + · · · + λ r d g r d ; z ) . If we have the trivial partition, then clearly we have h ( g ; ∅ ) = h ( g ) . We have the following lemma.
Lemma 3.4 (Lemma 2, [3]) . Let g = { g , . . . , g r d } ⊆ Q [ x , . . . , x n ] be a system of forms of degree d , and suppose we have a partition of variables x = ( y , z ) . Let y ′ be a distinct set of variables withthe same number of variables as y . Then we have h ( g ( y , z ) , g ( y ′ , z ); z ) = h ( g ; z ) . STANLEY YAO XIAO AND SHUNTARO YAMAGISHI
Given a system of forms which may not be regular, we want to obtain a regular system in acontrolled manner. The process in the following proposition is referred to as the regularization ofsystems in [3], and it is a crucial component of their method. Given a system of rational forms f ,via the regularization process we obtain another system R ( f ) which is regular, the number of formsit contains is controlled, and its level sets partition the level sets of f . We remark that condition(3) of Proposition 3.5, with a suitable choice of F , together with Corollary 3.3 implies that theresulting system is regular. Proposition 3.5 (Propositions 1 and 1’, [3]) . Let d > , and let F be any collection of non-decreasing functions F i : Z ≥ → Z ≥ (2 ≤ i ≤ d ) . For a collection of non-negative integers r , . . . , r d , there exist constants C ( r , . . . , r d , F ) , . . . , C d ( r , . . . , r d , F ) such that the following holds.Given a system of integral forms f = ( f ( d ) , . . . , f (1) ) ⊆ Z [ x , . . . , x n ] , where each f ( i ) is a sys-tem of r i forms of degree i (1 ≤ i ≤ d ) , and a partition of variables x = ( y , z ) , there exists asystem of forms R ( f ) = ( a ( d ) , . . . , a (1) ) satisfying the following. Let r ′ i = | a ( i ) | (1 ≤ i ≤ d ) , and R ′ = r ′ + · · · + r ′ d .(1) Each form of the system f can be written as a rational polynomial expression in the forms ofthe system R ( f ) . In particular, the level sets of R ( f ) partition those of f .(2) For each ≤ i ≤ d , r ′ i is at most C i ( r , . . . , r d , F ) .(3) The subsystem ( a ( d ) , . . . , a (2) ) satisfies h ( a ( i ) ) ≥ F i ( R ′ ) for each ≤ i ≤ d . Moreover, thelinear forms of subsystem a (1) are linearly independent over Q .(4) Let a ( i ) be the system obtained by removing from a ( i ) all forms that depend only on the z variables (2 ≤ i ≤ d ) . Then the subsystem ( a ( d ) , . . . , a (2) ) satisfies h ( a ( i ) ; z ) ≥ F i ( R ′ ) for each ≤ i ≤ d . We will be utilizing this proposition in Section 5 to obtain the minor arc estimate.4.
Technical Estimates
In this section, we provide results from [16] related to Weyl differencing that are necessary inobtaining estimates for the singular series in Section 6.1. The work here is similar to that of [16],which is in terms of forms instead of polynomials as in this section. It is stated in [16] with someexplanation that similar results for polynomials also follow, but the details are not shown. We choseto present the necessary details in order to make certain dependency of the constants explicit, whichare crucial in our estimates. Let us denote B = [ − , n . We shall refer to B ⊆ R n as a box, if B is of the form B = I × · · · × I n , where each I j is a closed or open or half open/closed interval (1 ≤ j ≤ n ). Given a function G ( x ),we define Γ d,G ( x , . . . , x d ) = X t =0 · · · X t d =0 ( − t + ··· + t d G ( t x + · · · + t d x d ) . Then it follows that Γ d,G is symmetric in its d arguments, and that Γ d,G ( x , . . . , x d − , ) = 0 [16,Section 11]. It is clear from the definition that Γ d,G + Γ d,G ′ = Γ d,G + G ′ . We also have that if G is aform of degree j , where d > j >
0, then Γ d,G = 0 [16, Lemma 11.2].
For α ∈ R , let k α k denote the distance from α to the closest integer. Given α = ( α , . . . , α n ) ∈ R n , we let k α k = max ≤ i ≤ n k α i k . Lemma 4.1. [16, Lemma 13.1]
Suppose G ( x ) = G (0) + G (1) ( x ) + · · · + G ( d ) ( x ) , where G ( j ) is aform of degree j with real coefficients (1 ≤ j ≤ d ) , and G (0) ∈ R . Let B be a box with sides ≤ ,let P > , and put S ′ = S ′ ( G, P, B ) = X x ∈ P B ∩ Z n e ( G ( x )) . Let e , . . . , e n be the standard basis vectors of R n . Then for any ε > , we have | S ′ | d − ≪ P (2 d − − d ) n + ε X n Y i =1 min( P, k Γ d,G ( d ) ( x , . . . , x d − , e i ) k − ) ! , where the sum P is over ( d − -tuples of integer points x , . . . , x d − in P B , and the implicitconstant in ≪ depends only on n, d, and ε . Lemma 4.2. [16, Lemma 14.2]
Make all the assumptions of Lemma 4.1. Suppose further that | S ′ | ≥ P n − Q where Q > . Let < η ≤ . Then the number N ( η ) of integral ( d − -tuples x , . . . , x d − ∈ P η B with k Γ d,G ( d ) ( x , . . . , x d − , e i ) k < P − d +( d − η ( i = 1 , . . . , n ) satisfies N ( η ) ≫ P n ( d − η − d − Q − ε , where the implicit constant in ≫ depends only on n, d, η, and ε . Let ψ = { ψ , . . . , ψ r d } be a system of rational polynomials of degree d . Let f = { f , . . . , f r d } bethe system of forms, where f i is the degree d portion of ψ i (1 ≤ i ≤ r d ). We define the followingexponential sum associated to ψ and B ,(4.1) S ( α ) = S ( ψ , B ; α ) = X x ∈ P B ∩ Z n e ( α · ψ ( x )) . Let e , . . . , e n be the standard basis vectors of C n . We define M d = M d ( f ) to be the set of( d − x , . . . , x d − ) ∈ ( C n ) d − for which the matrix[ m ij ] = [Γ d,f j ( x , . . . , x d − , e i )] (1 ≤ j ≤ r d , ≤ i ≤ n )has rank strictly less than r d . For R >
0, we denote z R ( M d ) to be the number of integer points( x , . . . , x d − ) on M d such that max ≤ i ≤ d − max ≤ j ≤ n | x ij | ≤ R, where x i = ( x i , . . . , x in ) (1 ≤ i ≤ d − P > Q >
0, and ε > d >
1. We then have:
Lemma 4.3. [16, Lemma 15.1]
Given a box B with sides ≤ , define the sum S ( α ) associated with ψ and B as in (4.1). Given < η ≤ , one of the following three alternatives must hold:(i) | S ( α ) | ≤ P n − Q .(ii) there exists n ∈ N such that n ≪ P r d ( d − η and k n α k ≪ P − d + r d ( d − η . (iii) z R ( M d ) ≫ R ( d − n − d − ( Q/η ) − ε holds with R = P η .All implicit constants depend at most on n, d, r d , η, ε and f .Proof. Take α ∈ R r d . Let α · ψ ( x ) = G (0) + G (1) ( x ) + · · · + G ( d ) ( x ), where G ( j ) is a form of degree j (1 ≤ j ≤ d ), and G (0) ∈ R . Suppose ( i ) fails, then we may apply Lemma 4.2. The number N ( η )of integral ( d − x , . . . , x d − in P η B with(4.2) k Γ d,G ( d ) ( x , . . . , x d − , e i ) k < P − d +( d − η ( i = 1 , . . . , n )satisfies N ( η ) ≫ R n ( d − − d − ( Q/η ) − ε , where R = P η , and the implicit constant in ≫ depends only on n, d, η, and ε .Recall ψ = { ψ , . . . , ψ r d } . Given x , . . . , x d − as above, we form the matrix[ m ij ] x ,..., x d − = [Γ d,ψ j ( x , . . . , x d − , e i )] (1 ≤ i ≤ n, ≤ j ≤ r d ) . Recall f j is the degree d portion of ψ j (1 ≤ j ≤ r d ), and f = { f , . . . , f r d } . Since each ψ j is ofdegree d , it follows that Γ d,ψ j = Γ d,f j (1 ≤ j ≤ r d ). It is also clear that G ( d ) ( x ) = α · f ( x ). Nowif this matrix [ m ij ] x ,..., x d − has rank less than r d for each of the ( d − N ( η ),then by the definition of z R ( M d ) we have that z R ( M d ) ≥ N ( η ) ≫ R n ( d − − d − ( Q/η ) − ε , where again the implicit constant in ≫ depends only on n, d, η, and ε . Thus we have ( iii ) in thiscase. Hence, we may suppose that at least one of these matrices, which we denote by [ m ij ], has rank r d . Without loss of generality, suppose the submatrix M formed by taking the first r d columns of[ m ij ] has rank r d . Let n = det( M ).It follows from the definition of Γ d,f j that every monomial occurring in Γ d,f j ( x , . . . , x d ) has somecomponent of x i as a factor for each 1 ≤ i ≤ d [16, Proof of Lemma 11.2]. Also, the maximumabsolute value of all coefficients of Γ d,f j is bounded by a constant dependent only on d and thecoefficients of f j [16, Lemma 11.3]. Therefore, by the construction of [ m ij ] we have m ij ≪ R d − , and hence n ≪ R r d ( d − = P r d ( d − η , where the implicit constants in ≪ depend only on r d and f .We have Γ d,G ( d ) = r d X j =1 Γ d,α j f j = r d X j =1 α j Γ d,f j . Hence, from (4.2) we may write r d X j =1 α j m ij = c i + β i (1 ≤ i ≤ n ) , where the c i are integers and the β i are real numbers bounded by the right hand side of (4.2). Let u , . . . , u r d be the solution of the system of linear equations(4.3) r d X j =1 u j m ij = n c i (1 ≤ i ≤ r d ) . Then(4.4) r d X j =1 ( n α j − u j ) m ij = n β i (1 ≤ i ≤ r d ) . By applying Cram´er’s rule to (4.3), it follows that the u j are integers. Also, by applying Cram´er’srule to (4.4), we obtain that k n α j k ≤ | n α j − u j | ≪ R ( d − r d − P − d +( d − η = P − d +( d − r d η , (4.5)where the implicit constant in ≪ depends only on r d and f . This completes the proof of Lemma4.3 (cid:3) We define g d ( f ) to be the largest real number such that(4.6) z P ( M d ) ≪ P n ( d − − g d ( f )+ ε holds for each ε >
0. It was proved in [16, pp. 280, Corollary] that(4.7) h ( f ) < d !(log 2) d ( g d ( f ) + ( d − r d ( r d − . Let γ d = 2 d − ( d − r d g d ( f )when g d ( f ) >
0. We let γ d = + ∞ if g d ( f ) = 0. We also define(4.8) γ ′ d = 2 d − g d ( f ) = γ d ( d − r d . Corollary 4.4. [16, pp.276, Corollary]
Given a box B with sides ≤ , we define the sum S ( α ) associated with ψ and B as in (4.1). Suppose ε ′ > is sufficiently small and Q > satisfies Qγ ′ d < . Then one of the following alternatives must hold:(i) | S ( α ) | ≤ P n − Q .(ii) there exists n ∈ N such that n ≪ P Qγ d + ε ′ and k n α k ≪ P − d + Qγ d + ε ′ , where the implicit constants in ≪ depend only on n, d, r d , ε ′ , Q and f . Note the fact that the implicit constant depends on f , but not on other lower order terms of ψ is an important feature which we make use of in Section 6.1. Proof.
Since Qγ ′ d <
1, we can choose ε > η = Qγ ′ d + ε satisfies0 < η ≤
1. Also with this choice of η , we have2 d − Qη = 2 d − QQγ ′ d + ε = g d ( f )1 + ε g d ( f ) / (2 d − Q ) < g d ( f ) . Then choose ε > d − Q/η + ε < g d ( f ). By the definition of g d ( f ) we have z R ( M d ) ≪ R n ( d − − d − Q/η − ε . Thus in this case we see that the statement ( iii ) in Lemma 4.3 can not occur with 0 < ε < ε . Alsothe equation η = Qγ ′ d + ε implies r d ( d − η = Qγ d + r d ( d − ε . Therefore, from Lemma 4.3 (applying it with 0 < ε < ε ) we obtain our result with ε ′ = r d ( d − ε . (cid:3) For the rest of this section, we assume ψ to be a system of integral polynomials of degree d .When the polynomials ψ in question are over Z , we consider the following.Hypothesis ( ⋆ ). Let B be a box in R n . For any ∆ >
0, there exists P = P ( f , Ω , ∆ , B ) suchthat for P > P , each α ∈ T r d satisfies at least one of the following two alternatives. Either(i) | S ( α ) | ≤ P n − ∆Ω , or(ii) there exists q = q ( α ) ∈ N such that q ≤ P ∆ and k q α k ≤ P − d +∆ . We will say that the restricted Hypothesis ( ⋆ ) holds if the above condition holds for each ∆ in0 < ∆ ≤ P in Hypothesis ( ⋆ ) only dependson f , and not on ψ . In other words, only the highest degree portion of the polynomials ψ play arole in this estimate. Proposition 4.5. [16, Proposition II ] Given a box B with sides ≤ , Hypothesis ( ⋆ ) is true forany Ω in (4.9) 0 < Ω < g d ( f )2 d − ( d − r d . Proof.
It follows from (4.9) that Ω γ d <
1. We set Q = ∆Ω, and let ε > Qγ d + ε < ∆. First, we suppose ∆ ≤ ( d − r d . In this case, it follows that Qγ ′ d <
1. Thus itfollows from Corollary 4.4 that there exists P = P ( f , Ω , ∆) such that whenever P > P , either( i ) | S ( α ) | ≤ P n − ∆Ω , or( ii ) there exists q ∈ N such that q ≤ P ∆ and k q α k ≤ P − d +∆ . On the other hand, if ∆ > ( d − r d , then the case ( ii ) above is always true by Dirichlet’s Theoremon Diophantine approximation. (cid:3) For each q ∈ N , we denote U q as the group of units in Z /q Z . Given m ∈ U r d q , we define(4.10) E ( q − m ) = E ( ψ , q ; q − m ) = q − n X x (mod q ) e ( q − m · ψ ( x )) . Lemma 4.6. [16, Lemma 7.1]
Suppose Ω satisfies (4.9). Then for < Q < Ω , we have (4.11) | E ( q − m ) | ≪ q − Q , where the implicit constant in ≪ depends only on f , Q and Ω . Again the fact that the implicit constant depends on f , but not on other lower order terms of ψ becomes crucial when we apply this lemma in Section 6.1. Proof.
Since E ( q − m ) = q − n S ( α ) with α = q − m , P = q and B = [0 , r d , and with our choice ofΩ we know that Hypothesis ( ⋆ ) is satisfied by Proposition 4.5. Thus we apply it with ∆ = Q/ Ω < q be sufficiently large, and suppose we are in case ( ii ) of Hypothesis ( ⋆ ). Then we know thereexists q ≤ q ∆ < q (when q = 1) with k q q − m k ≤ q − d +∆ < q − . Since ( m , q ) = 1, this is not possible. Therefore, we must have case ( i ) of Hypothesis ( ⋆ ), which isprecisely the inequality (4.11). (cid:3) Hardy-Littlewood Circle Method: Minor Arcs
For each q ∈ N , recall we let U q be the group of units in Z /q Z . When q = 1 we let U = { } .Let us denote T = R / Z . For a given value of C > ≤ q ≤ (log N ) C , we define the major arc M m,q ( C ) = { α ∈ T : k α − m/q k ≤ N − d (log N ) C } for each m ∈ U q . Recall k β k is the distance from β ∈ R to the nearest integer, which induces ametric on T via d ( α, β ) = k α − β k . These arcs are disjoint for N sufficiently large, and we define M ( C ) = [ q ≤ (log N ) C [ m ∈ U q M m,q ( C ) . We then define the minor arcs to be m ( C ) = T \ M ( C ) . We obtain the following bound on the minor arcs in this section.
Proposition 5.1.
Let b ( x ) ∈ Z [ x , . . . , x n ] be a polynomial of degree d . Let T ( b ; α ) be defined asin (1.2). Then there exists a positive number A d dependent only on d such that the following holds.Suppose b ( x ) satisfies h ⋆ ( f b ) > A d . Then, given any c > , there exists C > such that Z m ( C ) T ( b ; α ) dα ≪ N n − d (log N ) c . The proposition is achieved by splitting the exponential sum T ( b ; α ) over certain level sets basedon a decomposition of the polynomial b ( x ). Thus before we get into the proof of Proposition 5.1 wefirst establish this decomposition in six steps, where the resulting decomposition is given in (5.15).For simplicity, we let f ( x ) be the degree d portion of b ( x ) for the remainder of the paper. We let h = h ( f ), and let 0 < M < h ⋆ ( f ) ≤ h to be chosen later. Step 1: Decomposition of the variables.
As explained in the paragraph before (2.6), byrelabeling the variables if necessary we have f = ( x + ℓ ) v ′ + · · · + ( x M + ℓ M ) v ′ M + u ′ M +1 v ′ M +1 + · · · + u ′ h v ′ h , where each ℓ i is a linear form in Q [ x M +1 , . . . , x n ] (1 ≤ i ≤ M ), and each u ′ i ′ and v ′ j are rational formsof positive degree ( M + 1 ≤ i ′ ≤ h, ≤ j ≤ h ). We can then find a monomial x i x i · · · x i d , where M < i ≤ i ≤ · · · ≤ i d , of f with a non-zero coefficient. This is the case, for otherwise it meansthat every monomial of f is divisible by one of x , . . . , x M , and consequently that h = h ( f ) ≤ M ,which is a contradiction. We denote the distinct variables of { x i , x i , . . . , x i d } ⊆ { x M +1 , . . . , x n } by { w , . . . , w K } , and let w = ( w , . . . , w K ). Clearly, we have K ≤ d . We selected these K variablesfor the purpose of applying Weyl differencing later. We also label y = ( x , . . . , x M ) = ( y , . . . , y M )for notational convenience, let z = { x M +1 , . . . , x n }\ w , and denote z = ( z , . . . , z n − M − K ). We notethat each ℓ i is a rational linear form only in the w and the z variables (1 ≤ i ≤ h ). Step 2: Decomposition of f ( x ) . We define g M with respect to f as in (2.7). By Lemma 2.3,we have(5.1) f ( x ) = f ( w , y , z ) = g M ( w , y , z ) + f ( w , ( − ℓ , . . . , − ℓ M ) , z ) , where(5.2) h ( g M ( w , y , z )) ≥ M and h ( f ( w , ( − ℓ , . . . , − ℓ M ) , z )) = h − M. We then have(5.3) f ( , y , z ) = g M ( , y , z ) + f ( , ( − ℓ | w = , . . . , − ℓ M | w = ) , z ) . Let us denote f M ( z ) = f ( , ( − ℓ | w = , . . . , − ℓ M | w = ) , z ) . Consequently, we obtain from Lemma 2.1 and (5.2) that(5.4) h ( g M ( , y , z )) ≥ M − K ≥ M − d, and(5.5) h ( f M ( z )) ≥ h − M − K ≥ h − M − d. Step 3: Decomposition of b ( x ) with respect to w, y, and z. Let b M ( z ) = b ( , ( − ℓ | w = , . . . , − ℓ M | w = ) , z ) . It is clear that the degree d portion of the polynomial b ( , y , ) is g M ( , y , ). Let use denote b ( , y , z ) − b M ( z )(5.6) = d − X j =1 X ≤ t ≤···≤ t j ≤ M d − j X k =0 Ψ ( k ) t ,...,t j ( z ) ! y t · · · y t j + d X k =1 Ψ ( k ) ∅ ( z ) ! + g M ( , y , ) , where Ψ ( k ) t ,...,t j ( z ) and Ψ ( k ) ∅ ( z ) are forms of degree k . With these notations, we have the followingdecomposition, b ( w , y , z )(5.7) = b ( w , , ) + d − X j =1 X ≤ i ≤···≤ i j ≤ K d − j X k =1 Φ ( k ) i ,...,i j ( y , z ) ! w i · · · w i j + d − X j =1 X ≤ t ≤···≤ t j ≤ M d − j X k =0 Ψ ( k ) t ,...,t j ( z ) ! y t · · · y t j + d X k =1 Ψ ( k ) ∅ ( z ) ! + g M ( , y , )+ b M ( z ) − b ( , , ) , which we describe below. We note that Φ ( k ) i ,...,i j ( y , z ) are forms of degree k . The above decomposi-tion establishes the following. The term b ( w , , ) + d − X j =1 X ≤ i ≤···≤ i j ≤ K d − j X k =1 Φ ( k ) i ,...,i j ( y , z ) ! w i · · · w i j consists of all the monomials of b ( x ), which involve any variables of w . Consequently, we have b ( , y , z ) = d − X j =1 X ≤ t ≤···≤ t j ≤ M d − j X k =0 Ψ ( k ) t ,...,t j ( z ) ! y t · · · y t j + d X k =1 Ψ ( k ) ∅ ( z ) ! + g M ( , y , )+ b M ( z ) , and the degree d portion of b ( , y , z ) = f ( , y , z ). Clearly, the degree d portion of b M ( z ) is f M ( z ) = f ( , ( − ℓ | w = , . . . , − ℓ M | w = ) , z ) . It then follows from (5.3) and (5.6) that the degree d portion of d − X j =1 X ≤ t ≤···≤ t j ≤ M d − j X k =0 Ψ ( k ) t ,...,t j ( z ) ! y t · · · y t j + d X k =1 Ψ ( k ) ∅ ( z ) ! + g M ( , y , )is g M ( , y , z ) = d − X j =1 X ≤ t ≤···≤ t j ≤ M Ψ ( d − j ) t ,...,t j ( z ) y t · · · y t j + Ψ ( d ) ∅ ( z ) + g M ( , y , ) . We also know from (5.3) that g M ( , ( − ℓ | w = , . . . , − ℓ M | w = ) , z ) = 0, and consequently,Ψ ( d ) ∅ ( z ) = − d − X j =1 X ≤ t ≤···≤ t j ≤ M Ψ ( d − j ) t ,...,t j ( z ) y t · · · y t j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y i = − ℓ i | w = (1 ≤ i ≤ M ) − g M ( , ( − ℓ | w = , . . . , − ℓ M | w = ) , ) . In other words, Ψ ( d ) ∅ ( z ) can be expressed as a rational polynomial in the forms { Ψ ( d − j ) t ,...,t j ( z ) : 1 ≤ j ≤ d − , ≤ t ≤ · · · ≤ t j ≤ M } ∪ { ℓ i | w = : 1 ≤ i ≤ M } . Step 4: Regularization of systems Φ and Ψ . We denote by Φ = { Φ ( k ) i ,...,i j : 1 ≤ j ≤ d − , ≤ i ≤ · · · ≤ i j ≤ K, ≤ k ≤ d − j } . Note every polynomial of Φ has degree strictly lessthan d , and involves only the y and the z variables. Clearly, we have | Φ | ≤ d K d ≤ d d +2 . Weapply Proposition 3.5 to the system Φ with respect to the functions F = {F , . . . , F d − } , where F i ( t ) = ρ d,i (2 + 2 t ) + 2 t for 2 ≤ i ≤ d −
1, and obtain R (Φ) = ( a ( d − , . . . , a (1) ). For each form a ( s ) i ∈ a ( s ) (1 ≤ s ≤ d − , ≤ i ≤ | a ( s ) | ), we write(5.8) a ( s ) i ( y , z ) = s X k =0 X ≤ i ≤···≤ i k ≤ M e Ψ ( s − k ) s : i : i ,...,i k ( z ) y i · · · y i k , where each e Ψ ( s − k ) s : i : i ,...,i k ( z ) is a form of degree s − k . Thus each form a ( s ) i introduces at most ( s +1) M s ≤ dM d forms in z . Also for each 1 ≤ i ≤ d −
1, we denote a ( i ) to be the system obtained byremoving all forms that depend only on the z variables from a ( i ) . Let R (Φ) = ( a ( d − , . . . , a (1) ), R = P d − i =1 | a ( i ) | , and D = P d − i =1 i | a ( i ) | . By relabeling if necessary, we denote the elements of a ( s ) by a ( s ) = { a ( s ) i : 1 ≤ i ≤ | a ( s ) |} for each 1 ≤ s ≤ d − { Ψ ( k ) t ,...,t j ( z ) : 1 ≤ j ≤ d − , ≤ t ≤ · · · ≤ t j ≤ M, ≤ k ≤ d − j }∪ { Ψ ( k ) ∅ ( z ) : 1 ≤ k < d }∪ { ℓ i | w = : 1 ≤ i ≤ M }∪ { e Ψ ( s − k ) s : i : i ,...,i k ( z ) : 1 ≤ s ≤ d − , ≤ i ≤ | a ( s ) | , ≤ k ≤ s, ≤ i ≤ · · · ≤ i k ≤ M } . In other words, Ψ is the collection of ℓ i | w = , and all Ψ ( k ) t ,...,t j ( z ), e Ψ ( s − k ) s : i : i ,...,i k ( z ), and Ψ ( k ) ∅ ( z ) exceptΨ ( d ) ∅ ( z ). In particular, every polynomial of Ψ has degree strictly less than d . We can see that | Ψ | ≤ d M d + d + M + |R (Φ) | dM d . We let R (Ψ) be a regularization of Ψ with respect to the functions F = {F , . . . , F d − } , whereagain F i ( t ) = ρ d,i (2 + 2 t ) + 2 t for 2 ≤ i ≤ d −
1. Let us denote R (Ψ) = ( v ( d − , . . . , v (1) ), R = P d − i =1 | v ( i ) | , and D = P d − i =1 i | v ( i ) | .Let R ( i ) (Φ), Φ ( i ) , and R ( i ) (Ψ) denote the degree i forms of R (Φ), Φ, and R (Ψ), respectively.From Proposition 3.5, we know that each |R ( i ) (Φ) | = | a ( i ) | (1 ≤ i ≤ d − R ,is bounded by some constant dependent only on F , and | Φ ( d − | , . . . , | Φ (1) | . Thus we see that R is bounded by a constant dependent only on d . We set M = ρ d,d (2 + 2 R ) + 2 R + d, and note that M is bounded by a constant dependent only on d . By Proposition 3.5 again, we havethat each |R ( i ) (Ψ) | = | v ( i ) | (1 ≤ i ≤ d − R , is bounded by some constantdependent only on d , F , M , and | Φ ( d − | , . . . , | Φ (1) | . Thus R is bounded by a constant dependent only on d as well.We define(5.9) A d = max { ρ d,d (2 + 2 R ) + 4 R + 2 d, ρ d,d (2 + 2 R ) + 4 R + 2 d, · d − · ( d − · d !(log 2) d + 5 d } , and suppose h ⋆ ( f ) ≥ A d . We note that the third term inside the maximum function above is notrequired in this section, but this lower bound on A d becomes necessary in Section 6. With thischoice of A d , we have from (5.4) and (5.5) that(5.10) h ( f M ( z )) ≥ h − M − d ≥ ρ d,d (2 + 2 R ) + 2 R , and(5.11) h ( g M ( , y , z )) ≥ M − d ≥ ρ d,d (2 + 2 R ) + 2 R . Step 5: Definition of the level sets Z ( H ) and Y ( G ; H ) . For each H ∈ Z R , we define thefollowing set Z ( H ) = { z ∈ [0 , N ] n − M − K ∩ Z n − M − K : R (Ψ)( z ) = H } . By Proposition 3.5, we know that each of the polynomials Ψ ( k ) t ,...,t j ( z ) and Ψ ( k ) ∅ ( z ) in (5.7) can beexpressed as a rational polynomial in the forms of R (Ψ). Let us denoteΨ ( k ) t ,...,t j ( z ) = ˆ c ( k ) t ,...,t j ( R (Ψ)) and Ψ ( k ) ∅ ( z ) = ˆ c ( k ) ∅ ( R (Ψ)) , where ˆ c ( k ) t ,...,t j and ˆ c ( k ) ∅ are rational polynomials in R variables. Therefore, for any z ∈ Z ( H ), wehave Ψ ( k ) t ,...,t j ( z ) = ˆ c ( k ) t ,...,t j ( H ) and Ψ ( k ) ∅ ( z ) = ˆ c ( k ) ∅ ( H ) . Since each of the forms e Ψ ( s − k ) s : i : i ,...,i k ( z ) in (5.8) can be expressed as a rational polynomial in theforms of R (Ψ), let us denote e Ψ ( s − k ) s : i : i ,...,i k ( z ) = e c ( s − k ) s : i : i ,...,i k ( R (Ψ)) , where each e c ( s − k ) s : i : i ,...,i k is a rational polynomial in R variables. Therefore, for each a ( s ) i ∈ R (Φ) =( a ( d − , . . . , a (1) ), where 1 ≤ s ≤ d − ≤ i ≤ | a ( s ) | , we can write(5.12) a ( s ) i ( y , z ) = s X k =0 X ≤ i ≤···≤ i k ≤ M e c ( s − k ) s : i : i ,...,i k ( R (Ψ)) y i · · · y i k . Consequently, we can define the following polynomial for each 1 ≤ s ≤ d − ≤ i ≤ | a ( s ) | ,(5.13) a ( s ) i ( y , Z ( H )) = s X k =0 X ≤ i ≤···≤ i k ≤ M e c ( s − k ) s : i : i ,...,i k ( H ) y i · · · y i k , so that given any z ∈ Z ( H ), we have a ( s ) i ( y , z ) = a ( s ) i ( y , Z ( H )) . We also define R (Φ)( y , Z ( H )) = { a ( s ) i ( y , Z ( H )) : 1 ≤ s ≤ d − , ≤ i ≤ | a ( s ) |} , which consists of R polynomials with possible repetitions. For each G ∈ Z R , we let Y ( G ; H ) = { y ∈ [0 , N ] M ∩ Z M : R (Φ)( y , Z ( H )) = G } . Step 6: Decomposition of b ( w , y , z ) when ( y , z ) ∈ Y ( G ; H ) × Z ( H ) . Recall Φ is the collectionof all Φ ( k ) i ,...,i j ( y , z ) in (5.7), and that each Φ ( k ) i ,...,i j ( y , z ) can be expressed as a rational polynomialin the forms of R (Φ). Thus, it follows from this fact and (5.13) that each Φ ( k ) i ,...,i j ( y , z ) is constanton ( y , z ) ∈ Y ( G ; H ) × Z ( H ), and we denote this constant value by c ( k ) i ,...,i j ( G , H ). Therefore, forany choice of z ∈ Z ( H ) and y ∈ Y ( G ; H ), the polynomial b ( x ) takes the following shape b ( w , y , z )(5.14) = b ( w , , ) + d − X j =1 X ≤ i ≤···≤ i j ≤ K d − j X k =1 c ( k ) i ,...,i j ( G , H ) ! w i · · · w i j + d − X j =1 X ≤ t ≤···≤ t j ≤ M d − j X k =0 ˆ c ( k ) t ,...,t j ( H ) ! y t · · · y t j + d X k =1 ˆ c ( k ) ∅ ( H ) ! + g M ( , y , )+ b M ( z ) − b ( , , ) . We label C ( w , G , H ) = b ( w , , ) + d − X j =1 X ≤ i ≤···≤ i j ≤ K d − j X k =1 c ( k ) i ,...,i j ( G , H ) ! w i · · · w i j , and C ( y , H ) = d − X j =1 X ≤ t ≤···≤ t j ≤ M d − j X k =0 ˆ c ( k ) t ,...,t j ( H ) ! y t · · · y t j + d X k =1 ˆ c ( k ) ∅ ( H ) ! + g M ( , y , ) , so that for z ∈ Z ( H ) and y ∈ Y ( G ; H ), we have(5.15) b ( w , y , z ) = C ( w , G , H ) + C ( y , H ) + b M ( z ) − b ( , , ) . Proof of Proposition 5.1.
We are now in position to prove Proposition 5.1.
Proof.
Using the notations above we define the following three exponential sums, S ( α, G , H ) = X w ∈ [0 ,N ] K ∩ Z K Λ( w ) e ( α · C ( w , G , H )) ,S ( α, G , H ) = X y ∈ Y ( G ; H ) Λ( y ) e ( α · C ( y , H )) , and S ( α, H ) = X z ∈ Z ( H ) Λ( z ) e ( α · b M ( z ) − α · b ( , , )) . Let L ( N ) = { H ∈ Z R : Z ( H ) = ∅} , and for each H ∈ L ( N ), let L ( N ; H ) = { G ∈ Z R : Y ( G , H ) = ∅} . It then follows that(5.16) |L ( N ) | ≪ N D and |L ( N ; H ) | ≪ N D , where the implicit constant in the second inequality is independent of H . In order to prove the firstinequality, let C be the largest absolute value of all coefficients of the polynomials in R (Ψ). Also let M be the largest number of monomials with non-zero coefficients in any of the polynomials in R (Ψ). Then we have |L ( N ) | ≤ (2 C · M ) R · ( N + 1) D . To see the second inequality, we let C ′ be the largest absolute value of all coefficients of thepolynomials a ( s ) i ( y , z ) in R (Φ), and let M ′ be the largest number of monomials with non-zerocoefficients in any of these polynomials. Then we see that the number of values taken by a ( s ) i ( y , z )as ( y , z ) varies in [0 , N ] n − K is ≤ (2 C ′ · M ′ ) · ( N + 1) s . Therefore, we have L ( N ; H ) = { G ∈ Z R : Y ( G , H ) = ∅} = { G ∈ Z R : ∃ y ∈ [0 , N ] M ∩ Z M , R (Φ)( y , Z ( H )) = G }⊆ { G ∈ Z R : ∃ ( y , z ) ∈ [0 , N ] n − K ∩ Z n − K , R (Φ)( y , z ) = G } , and the cardinality of the last set is ≤ (2 C ′ · M ′ ) R · ( N + 1) D . By the Cauchy-Schwarz inequality and (5.16), we obtain (cid:12)(cid:12)(cid:12) Z m ( C ) T ( b ; α ) dα (cid:12)(cid:12)(cid:12) (5.17) ≤ (cid:12)(cid:12)(cid:12) X H ∈L ( N ) X G ∈L ( N ; H ) Z m ( C ) X w ∈ [0 ,N ] K ∩ Z K z ∈ Z ( H ) y ∈ Y ( G ; H ) Λ( w )Λ( y )Λ( z ) · e ( α · ( C ( w , G , H ) + C ( y , H ) + b M ( z ) − b ( , , ) )) dα (cid:12)(cid:12)(cid:12) ≪ N D + D X H ∈L ( N ) X G ∈L ( N ; H ) (cid:12)(cid:12)(cid:12) Z m ( C ) S ( α, G , H ) S ( α, G , H ) S ( α, H ) dα (cid:12)(cid:12)(cid:12) ≪ N D + D sup H ∈L ( N ) G ∈L ( N ; H ) sup α ∈ m ( C ) | S ( α, G , H ) | · X H ∈L ( N ) X G ∈L ( N ; H ) k S ( · , G , H ) k k S ( · , H ) k , where k · k denotes the L -norm on [0 , k S ( · , G , H ) k k S ( · , H ) k ≤ (log N ) n − K N ( G ; H ) N ( H ) , where N ( G ; H ) = |{ ( y , y ′ ) ∈ Y ( G ; H ) × Y ( G ; H ) : C ( y , H ) = C ( y ′ , H ) }| , and N ( H ) = |{ ( z , z ′ ) ∈ Z ( H ) × Z ( H ) : b M ( z ) = b M ( z ′ ) }| . With these notations, we may further bound (5.17) as follows (cid:12)(cid:12)(cid:12) Z m ( C ) T ( b ; α ) dα (cid:12)(cid:12)(cid:12) ≪ (log N ) n − K N D + D sup H ∈L ( N ) G ∈L ( N ; H ) sup α ∈ m ( C ) | S ( α, G , H ) | W , where W = X H ∈L ( N ) X G ∈L ( N ; H ) N ( G ; H ) N ( H ) . We can express W as the number of solutions y , y ′ ∈ [0 , N ] M ∩ Z M and z , z ′ ∈ [0 , N ] n − M − K ∩ Z n − M − K of the system R (Ψ)( z ) = R (Ψ)( z ′ ) = H (5.18) R (Φ)( y , Z ( H )) = R (Φ)( y ′ , Z ( H )) = G C ( y , H ) = C ( y ′ , H ) b M ( z ) = b M ( z ′ )for any H ∈ L ( N ) and G ∈ L ( N ; H ). We know that the system R (Φ)( y , Z ( H )) is identical to R (Φ)( y , z ) for any choice of z ∈ Z ( H ) and any y ∈ [0 , N ] M ∩ Z M . Similarly, we know that thepolynomial C ( y , H ) is identical to b ( , y , z ) − b M ( z ) for any choice of z ∈ Z ( H ). Therefore,since R (Ψ)( z ) = H implies that z ∈ Z ( H ), we can rearrange the system (5.18) and deduce that W is the number of solutions y , y ′ ∈ [0 , N ] M ∩ Z M and z , z ′ ∈ [0 , N ] n − M − K ∩ Z n − M − K of the followingsystem R (Ψ)( z ) = R (Ψ)( z ′ )(5.19) R (Φ)( y , z ) = R (Φ)( y ′ , z ) b ( , y , z ) − b M ( z ) = b ( , y ′ , z ) − b M ( z ) b M ( z ) = b M ( z ′ ) . Our result follows from the following two claims.Claim 1: Given any c >
0, there exists
C > H ∈L ( N ) G ∈L ( N ; H ) sup α ∈ m ( C ) | S ( α, G , H ) | ≪ N K (log N ) c . Claim 2: We have the following bound on W , W ≪ N n − K − d − D − D . By substituting the bounds from the above two claims into (5.18), we obtain for any c >
C > Z m ( C ) T ( b ; α ) dα ≪ N n − d (log N ) c , and this completes the proof of our proposition. Therefore, we only need to establish Claims 1 and2. Claim 1 is obtained via Weyl differencing. Since the set up for our Claim 1 is the same as thatof [3], we omit the proof of Claim 1 and refer the reader to [3, pp. 725].We now present the proof of Claim 2. From (5.19), we can write W = X z ∈ [0 ,N ] n − M − K ∩ Z n − M − K T ( z ) · T ( z ) , where T ( z ) is the number of solutions y , y ′ ∈ [0 , N ] M ∩ Z M to the system b ( , y , z ) = b ( , y ′ , z ) R (Φ)( y , z ) = R (Φ)( y ′ , z ) , and T ( z ) is the number of solutions z ′ ∈ [0 , N ] n − M − K ∩ Z n − M − K to the system b M ( z ) = b M ( z ′ ) R (Ψ)( z ) = R (Ψ)( z ′ ) . Define W i = P z T i ( z ) ( i = 1 ,
2) so that we have W ≤ W W by the Cauchy-Schwarz inequality.We first estimate W , which we can deduce to be the number of solutions y , y ′ , u , u ′ ∈ [0 , N ] M ∩ Z M and z ∈ [0 , N ] n − M − K ∩ Z n − M − K satisfying the equations b ( , y , z ) − b ( , y ′ , z ) = 0(5.20) b ( , u , z ) − b ( , u ′ , z ) = 0 R (Φ)( y , z ) − R (Φ)( y ′ , z ) = 0 R (Φ)( u , z ) − R (Φ)( u ′ , z ) = 0 . We consider the h -invariant of the system of forms on the left hand side of (5.20), and show that itis a regular system. The first two equations of (5.20) are the degree d polynomials of the system,and let h d be the h -invariant of these two polynomials. Suppose for some λ, µ ∈ Q , not both 0, wehave λ · ( f ( , y , z ) − f ( , y ′ , z )) + µ · ( f ( , u , z ) − f ( , u ′ , z )) = h d X j =1 U j · V j , where U j = U j ( y , y ′ , u , u ′ , z ) and V j = V j ( y , y ′ , u , u ′ , z ) are rational forms of positive degree(1 ≤ j ≤ h d ). Without loss of generality, suppose λ = 0. Let ℓ = ( − ℓ | w = , . . . , − ℓ M | w = ). If weset u = u ′ = y ′ = ℓ , then the above equation becomes g M ( , y , z ) = f ( , y , z ) − f M ( z ) = 1 λ h d X j =1 U j ( y , ℓ , ℓ , ℓ , z ) · V j ( y , ℓ , ℓ , ℓ , z ) . Therefore, we obtain from (5.11) h d ≥ h ( g M ( , y , z )) ≥ ρ d,d (2 + 2 R ) + 2 R ≥ ρ d,d (2 + 2 R − | a (1) | ) + 2 | a (1) | . For each 1 ≤ i ≤ d −
1, denote by R (Φ) ( i ) ( y , z ) − R (Φ) ( i ) ( y ′ , z ) = { a ( i ) j ( y , z ) − a ( i ) j ( y ′ , z ) : 1 ≤ j ≤ | a ( i ) |} , the system of degree i polynomials of R (Φ)( y , z ) − R (Φ)( y ′ , z ). We also define R (Φ) ( i ) ( u , z ) − R (Φ) ( i ) ( u ′ , z )in a similar manner. We apply Lemma 3.4 to estimate the h -invariant of the degree i forms of thesystem (5.20) for each 2 ≤ i ≤ d − h (cid:16) R (Φ) ( i ) ( y , z ) − R (Φ) ( i ) ( y ′ , z ) , R (Φ) ( i ) ( u , z ) − R (Φ) ( i ) ( u ′ , z ) (cid:17) ≥ h (cid:16) R (Φ) ( i ) ( y , z ) − R (Φ) ( i ) ( y ′ , z ) , R (Φ) ( i ) ( u , z ) − R (Φ) ( i ) ( u ′ , z ); z (cid:17) = h (cid:16) R (Φ) ( i ) ( y , z ) − R (Φ) ( i ) ( y ′ , z ); z (cid:17) ≥ h (cid:16) R (Φ) ( i ) ( y , z ) , R (Φ) ( i ) ( y ′ , z ); z (cid:17) ≥ h (cid:16) R (Φ) ( i ) ( y , z ); z (cid:17) ≥ ρ d,i (2 + 2 R ) + 2 R ≥ ρ d,i (2 + 2 R − | a (1) | ) + 2 | a (1) | . We also have to show that the linear forms of the system (5.20) are linearly independent over Q .Recall the linear forms of R (Φ) (1) ( y , z ) are linearly independent over Q , and do not include any linear forms that depend only on the z variables, and similarly for R (Φ) (1) ( y ′ , z ), R (Φ) (1) ( u , z ),and R (Φ) (1) ( u ′ , z ). We leave it as a basic exercise for the reader to verify that the linear forms of R (Φ) (1) ( y , z ) − R (Φ) (1) ( y ′ , z ) [ R (Φ) (1) ( u , z ) − R (Φ) (1) ( u ′ , z )are linearly independent over Q .Therefore, it follows from Corollary 3.3 that W ≪ N n +3 M − K − (2 d +2 D ) . We now estimate W , which we can deduce to be the number of solutions z , z ′ , z ′′ ∈ [0 , N ] n − M − K ∩ Z n − M − K satisfying the equations b M ( z ) − b M ( z ′ ) = 0(5.21) b M ( z ) − b M ( z ′′ ) = 0 R (Ψ)( z ) − R (Ψ)( z ′ ) = 0 R (Ψ)( z ) − R (Ψ)( z ′′ ) = 0 . We consider the h -invariant of the system of forms on the left hand side of (5.21), and show that itis a regular system. The first two equations of (5.21) are the degree d polynomials of the system,and let h d be the h -invariant of these two polynomials. Suppose for some λ, µ ∈ Q , not both 0, wehave λ · ( f M ( z ) − f M ( z ′ )) + µ · ( f M ( z ) − f M ( z ′′ )) = h d X j =1 U j · V j , where U j = U j ( z , z ′ , z ′′ ) and V j = V j ( z , z ′ , z ′′ ) are rational forms of positive degree (1 ≤ j ≤ h d ).We consider two cases, ( λ + µ ) = 0 and ( λ + µ ) = 0. Suppose ( λ + µ ) = 0. If we set z ′ = z ′′ = ,then the above equation becomes( λ + µ ) · f M ( z ) = h d X j =1 U j ( z , , ) · V j ( z , , ) . Thus we obtain h d ≥ h ( f M ( z )) . On the other hand, suppose ( λ + µ ) = 0, then the above equa-tion (5.22) simplifies to f M ( z ′ ) − f M ( z ′′ ) = − λ h d X j =1 U j · V j . From this equation, we substitute z ′′ = to obtain h d ≥ h ( f M ( z ′ )) . Therefore, in either case weobtain from (5.10) that h d ≥ h ( f M ( z )) ≥ ρ d,d (2 + 2 R ) + 2 R ≥ ρ d,d (2 + 2 R − | v (1) | ) + 2 | v (1) | . Recall we defined R (Ψ) = ( v ( d − , . . . , v (1) ), where v ( i ) = R ( i ) (Ψ) are the degree i forms of R (Ψ) (1 ≤ i ≤ d − ≤ i ≤ d −
1. Let m i = | v ( i ) | , and we label the forms of v ( i ) to be v ( i )1 , . . . , v ( i ) m i . Let h i be the h -invariant of the degree i forms of the system (5.21). Then forsome λ , µ ∈ Q m i , not both , we have(5.22) m i X j =1 λ j · ( v ( i ) j ( z ) − v ( i ) j ( z ′ )) + m i X j =1 µ j · ( v ( i ) j ( z ) − v ( i ) j ( z ′′ )) = h i X t =1 U t · V t , where U t = U t ( z , z ′ , z ′′ ) and V t = V t ( z , z ′ , z ′′ ) are forms of positive degree (1 ≤ t ≤ h i ). We considertwo cases, ( λ + µ ) = and ( λ + µ ) = . Suppose ( λ + µ ) = . In this case, we set z ′ = z ′′ = , and equation (5.22) simplifies to m i X j =1 ( λ j + µ j ) · v ( i ) j ( z ) = h i X t =1 U t ( z , , ) · V t ( z , , ) . Therefore, it follows that h i ≥ h ( v ( i ) ) ≥ ρ d,i (2 + 2 R ) + 2 R ≥ ρ d,i (2 + 2 R − | v ( i ) | ) + 2 | v (1) | . On the other hand, suppose ( λ + µ ) = . Then equation (5.22) simplifies to m i X j =1 − λ j · ( v ( i ) j ( z ′ ) − v ( i ) j ( z ′′ )) = h i X t =1 U t · V t . From this equation, we substitute z ′′ = to obtain h i ≥ h ( v ( i ) ) ≥ ρ d,i (2 + 2 R ) + 2 R ≥ ρ d,i (2 + 2 R − | v ( i ) | ) + 2 | v (1) | . We also have to show that the linear forms of the system (5.21),(5.23) { v (1) ( z ) − v (1) ( z ′ ) } ∪ { v (1) ( z ) − v (1) ( z ′′ ) } , are linearly independent over Q . Recall the linear forms of v (1) ( z ) are linearly independent over Q .The linear independence over Q of the system of linear forms (5.23) follows from this fact, and weleave the verification as a basic exercise for the reader.Therefore, we obtain by Corollary 3.3 that W ≪ N n − M − K ) − (2 d +2 D ) . Combining the bounds for W and W together, we obtain W ≤ W / W / ≪ N n − K − (2 d + D + D ) , which proves Claim 2. (cid:3) Hardy-Littlewood Circle Method: Major Arcs
Recall f ( x ) is the degree d portion of the degree d polynomial b ( x ) ∈ Z [ x , . . . , x n ]. In thissection we assume that f ( x ) satisfies h ( f ) > A d , where A d is defined in (5.9). We define g d ( f ) asin (4.6) with f = { f } and r d = 1. It then follows from (4.7) that A d < h ( f ) ≤ (log 2) − d · d ! · g d ( f ) . From this bound and our choice of A d in (5.9), we have(6.1) 2 d − g d ( f ) < d !2 d − (log 2) d A d < d !2 d − (log 2) d ( A d − d ) ≤ d − . We take Ω to be 4 < Ω < ≤ ( A d − d ) · (log 2) d d − ( d − d ! ≤ g d ( f )2 d − ( d − . Therefore, with this choice of Ω, we have that b ( x ) satisfies the Hypothesis ( ⋆ ) with B by Propo-sition 4.5. We then choose Q to satisfy 0 < Q < Ω and(6.2) Q · d − g d ( f ) < . In particular, we may choose Q to satisfy Q >
4. We fix these values of Ω and Q throughout thissection. We note that with these choices of Ω and Q , we have(6.3) 0 < Ω ≤ ( A d − dQ ) · (log 2) d d − ( d − d ! . The work of this section is based on [3] and it is similar to their treatment of the major arcs.However, we had to tailor their argument to be in terms of the h -invariant instead of the Birchrank.We define the following sums(6.4) e S m,q = X k ∈ U nq e ( b ( k ) · m/q ) , B ( q ) = X m ∈ U q φ ( q ) n e S m,q , and S ( N ) = X q ≤ (log N ) C B ( q ) , where φ is Euler’s totient function. Recall we denote B = [0 , n . We have the following estimateon the major arcs which is a consequence of [3, (6.1)] and [3, Lemma 6], and leave the details tothe reader. We remark that although it is assumed in [3, Lemma 6] that C is sufficiently large, itin fact follows from their proof that assuming C >
Lemma 6.1 (Lemma 6, [3]) . Let c > , C > , q ≤ (log N ) C , and m ∈ U q . Then we have Z M m,q ( C ) T ( b ; α ) dα = 1 φ ( q ) n e S m,q J + O (cid:18) N n − d (log N ) c (cid:19) , where J = Z | τ |≤ N − d (log N ) C Z u ∈ N B e ( τ b ( u )) du dτ. Note J is independent of m and q . We now simplify the expression for J . Let I ( η ) = Z B e ( ηf ( ξ )) d ξ For any ε >
0, the inner integral of J can be expressed as Z u ∈ N B e ( τ b ( u )) du = Z u ∈ N B e ( τ f ( u )) du + O ( N n − ε )= N n Z ξ ∈ B e ( N d τ f ( ξ )) d ξ + O ( N n − ε )= N n · I ( N d τ ) + O ( N n − ε ) , where we used the change of variable u = N ξ to obtain the second equality above.We define J ( L ) = Z | η |≤ L I ( η ) dη. Then we can simplify J as J = N n − d · J ((log N ) C ) + O ( N n − d − ε (log N ) C ) . Since we have Ω > ⋆ ), and in particular the restricted Hypothesis ( ⋆ ), itfollows by [16, Lemma 8.1] that(6.5) I ( η ) ≪ min(1 , | η | − ) . As stated in [16, Section 3], it follows from (6.5) that µ ( ∞ ) = Z R I ( η ) dη exists. Furthermore, we have(6.6) (cid:12)(cid:12)(cid:12) µ ( ∞ ) − J ( L ) (cid:12)(cid:12)(cid:12) ≪ L − . We also have µ ( ∞ ) > f ( x ) = 0 has a non-singular real solution in the interior of B = [0 , n (see [3, pp. 704]).Therefore, we obtain the following estimate as a consequence of the definition of the major arcsand Lemma 6.1. Lemma 6.2.
Suppose h ( f ) > A d , where we define A d as in (5.9). Then given any c > , thereexists C > such that we have Z M ( C ) T ( b ; α ) dα = S ( N ) µ ( ∞ ) N n − d + O (cid:18) S ( N ) N n − d (log N ) C + N n − d (log N ) c (cid:19) . Singular Series.
We obtain the following estimate on the exponential sum e S m,q definedin (6.4). Lemma 6.3.
Suppose h ( f ) > A d , where we define A d as in (5.9). Let p be a prime and let q = p t , t ∈ N . For m ∈ U q , we have the following bounds e S m,q ≪ (cid:26) q n − Q , if t ≤ d,p Q q n − Q , if t > d, where the implicit constants are independent of p .Proof. We consider the two cases t ≤ d and t > d separately. We apply the inclusion-exclusionprinciple to bound e S m,q when q = p t and t ≤ d , e S m,q = X k ∈ ( Z /q Z ) n n Y i =1 − X u i ∈ Z /p t − Z k i = pu i e ( b ( k ) · m/q )= X I ⊆{ , ,...,n } ( − | I | X u ∈ ( Z /p t − Z ) | I | X k ∈ ( Z /q Z ) n F I ( k ; u ) e ( b ( k ) · m/q ) , where k i = pu i denotes a characteristic function and F I ( k ; u ) = Y i ∈ I k i = pu i for u ∈ ( Z /p t − Z ) | I | . In other words, F I ( k ; u ) is the characteristic function of the set H I, u = { k ∈ ( Z /q Z ) n : k i = pu i ( i ∈ I ) } . We now bound the summand in the final expression of (6.7) by furtherconsidering two cases, | I | ≥ tQ and | I | < tQ . In the first case | I | ≥ tQ , we use the following trivialestimate (cid:12)(cid:12)(cid:12) X u ∈ ( Z /p t − Z ) | I | X k ∈ ( Z /q Z ) n F I ( k ; u ) e ( b ( k ) · m/q ) (cid:12)(cid:12)(cid:12) ≤ p ( t − | I | ( p t ) n −| I | = q n −| I | /t ≤ q n − Q . On the other hand, suppose | I | < tQ . Let g b ( x ) be the polynomial obtained by substituting x i = pu i ( i ∈ I ) to b ( x ). Thus g b ( x ) is a polynomial in n − | I | variables. We can also deduce easilythat the degree d portion of g b ( x ), which we denote f g b , is obtained by substituting x i = 0 ( i ∈ I )to the degree d portion of b ( x ). Hence, we have f g b = f | x i =0 ( i ∈ I ) . Consequently, we obtain by Lemma 2.1 that h ( f g b ) ≥ h ( f ) − | I | > h ( f ) − dQ > A d − dQ. By our choice of Q and Ω, and from (4.7) and (6.3), we have0 < Q < Ω < h ( f g b ) · (log 2) d d − ( d − d ! ≤ g d ( f g b )2 d − ( d − . Therefore, with these notations we have by Lemma 4.6 that X k ∈ ( Z /q Z ) n F I ( k ; u ) e ( b ( k ) · m/q ) = X s ∈ ( Z /q Z ) n −| I | e ( g b ( s ) · m/q ) = q n −| I | E ( g b , q ; m/q ) ≪ q n −| I |− Q . Thus, we obtain X u ∈ ( Z /p t − Z ) | I | X k ∈ ( Z /q Z ) n F I ( k ; u ) e ( b ( k ) · m/q ) ≪ ( p t − ) | I | q n −| I |− Q ≤ q n − Q . Consequently, combining the two cases | I | ≥ tQ and | I | < tQ together, we obtain e S m,q ≪ q n − Q when t ≤ d .We now consider the case q = p t when t > d . By the definition of e S m,q , we have e S m,q = X k ′ ∈ U np X s ∈ ( Z / ( p t − Z )) n e ( b ( k ′ + p s ) · m/q ) = X k ′ ∈ U np X s ∈ [0 ,p t − ) n e ( b ( k ′ + p s ) · m/q ) . For each fixed k ′ ∈ U np , we have b ( k ′ + p s ) = p d f ( s ) + χ p, k ′ ( s ) , where χ p, k ′ ( x ) is a polynomial of degree at most d − p and k ′ .We apply Corollary 4.4 with r d = 1, ψ ( x ) = f ( x ) + p d χ p, k ′ ( x ), α = m/p t − d , B = [0 , n , and P = p t − . Let ε ′ > Q > Q · d − g d ( f ) < . Let γ d and γ ′ d be as in the paragraph before Corollary 4.4 with f = { f } and r d = 1. Suppose thealternative ( ii ) of Corollary 4.4 holds. Then we know there exists n ∈ N such that n ≪ ( p t − − Qγ d + ε ′ and(6.7) k n ( m/p t − d ) k ≪ ( p t − − − d + Qγ d + ε ′ ≤ (cid:18) p t − (cid:19) − d + Qγ d + ε ′ . However, this is not possible once p t is sufficiently large with respect to n, d, ε ′ , Q , and f , for thefollowing reason. First note that n can not be divisible by p t − d for p t sufficiently large, because Qγ d + ε ′ < Qγ ′ d <
1. Since n ∈ N is not divisible by p t − d and ( m, p ) = 1, we have k n ( m/p t − d ) k ≥ p t − d , which contradicts (6.7) for p t sufficiently large. Thus by Corollary 4.4, we can bound the inner sumof (6.7) by X s ∈ [0 ,p t − ) n e (cid:18)(cid:18) f ( s ) + 1 p d χ p, k ′ ( s ) (cid:19) · m/p t − d (cid:19) ≪ ( p t − ) n − Q , where the implicit constant depends at most on n, d, ε ′ , Q , and f . Therefore, we can bound (6.7)as follows e S m,q ≤ X k ′ ∈ U np (cid:12)(cid:12)(cid:12) X s ∈ [0 ,p t − ) n e (cid:18)(cid:18) f ( s ) + 1 p d χ p, k ′ ( s ) (cid:19) · m/p t − d (cid:19) (cid:12)(cid:12)(cid:12) ≪ p n ( p t − ) n − Q = p Q q n − Q . (cid:3) For each prime p , we define(6.8) µ ( p ) = 1 + ∞ X t =1 B ( p t ) , which converges absolutely provided that h ( f ) > A d as we see in the following lemma. As statedin [3], by following the outline of L. K. Hua [11, Chapter VII, §
2, Lemma 8.1] one can show that B ( q ) is a multiplicative function of q . Therefore, we consider the following identity(6.9) S ( ∞ ) := lim N →∞ S ( N ) = Y p prime µ ( p ) . Lemma 6.4.
There exists δ > such that for each prime p , we have µ ( p ) = 1 + O ( p − − δ ) , where the implicit constant is independent of p . Furthermore, we have (cid:12)(cid:12)(cid:12) S ( N ) − S ( ∞ ) (cid:12)(cid:12)(cid:12) ≪ (log N ) − Cδ for some δ > . Therefore, the limit in (6.9) exists, and the product in (6.9) converges. We leave the details thatthese two quantities are equal to the reader.
Proof.
Recall our choice of Q satisfies Q >
4. Let ε > e Q = Q − ε > ≥ d/ ( d − Q = e Q + ε into the bounds in Lemma 6.3. It is then clearthat we may assume the implicit constant in Lemma 6.3 is 1 for p sufficiently large with the costof using e Q in place of Q . For any t ∈ N , we know that φ ( p t ) = p t (1 − /p ) ≥ p t . Therefore, byconsidering the two cases as in Lemma 6.3, we obtain | µ ( p ) − | ≪ X ≤ t ≤ d p t p − nt p nt − t e Q + X t>d p t p − nt p e Q + nt − t e Q ≪ p − e Q + p e Q p − ( d +1)( e Q − ≪ p − − δ , for some δ >
0. We note that the implicit constants in ≪ are independent of p here.Let q = p t · · · p t v v be the prime factorization of q ∈ N . Without loss of generality, suppose wehave t j ≤ d (1 ≤ j ≤ v ) and t j > d ( v < j ≤ v ). By the multiplicativity of B ( q ), it also followsfrom Lemma 6.3 that B ( q ) = B ( p t ) · · · B ( p t v v ) ≪ q − e Q · v Y j = v +1 p e Qj ≤ q − e Q · q e Q/d ≤ q − − δ , for some δ >
0. We note that the implicit constant in ≪ is independent of q here, because theimplicit constant in Lemma 6.3 is 1 for p sufficiently large as mentioned above. Therefore, weobtain (cid:12)(cid:12)(cid:12) S ( N ) − S ( ∞ ) (cid:12)(cid:12)(cid:12) ≤ X q> (log N ) C | B ( q ) | ≪ X q> (log N ) C q − − δ ≪ (log N ) − Cδ . (cid:3) Let ν t ( p ) denote the number of solutions x ∈ ( U p t ) n to the congruence b ( x ) ≡ p t ) . (6.10)It can be deduced that1 + t X j =1 B ( p j ) = 1 φ ( p t ) n X k ∈ ( U pt ) n X m ∈ Z / ( p t Z ) e (cid:0) b ( k ) · m/p t (cid:1) = p t φ ( p t ) n ν t ( p ) . Therefore, provided h ( b ) > A d we obtain µ ( p ) = lim t →∞ p t ν t ( p ) φ ( p t ) n . At this point we refer the reader to [3, pp. 704, 736] to conclude µ ( p ) > b ( x ) = 0has a non-singular solution in Z × p , the units of p -adic integers. It then follows from Lemma 6.4 thatif the equation b ( x ) = 0 has a non-singular solution in Z × p for every prime p , then Q p prime µ ( p ) > . Finally, we let C b = µ ( ∞ ) Q p prime µ ( p ) and Theorem 1.1 follows as a consequence of Lemmas 6.2and 6.4, and Proposition 5.1. References [1] B. J. Birch,
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