An explicit version of Birch s Theorem

Transcription

1 ACTA ARITHMETICA LXXXV.1 (1998) An explicit version of Birch s Theorem by Trevor D. Wooley (Ann Arbor, Mich.) 1. Introuction. Birch [1] has shown that a system of homogeneous polynomials with rational coefficients possesses a non-trivial rational zero provie only that these polynomials are of o egree, an the system has sufficiently many variables in terms of the number an egrees of these polynomials. While bouns have been obtaine for the number of variables which suffice to guarantee the existence of a non-trivial zero, in all but the simplest cases such bouns as are available are too weak to have warrante explicit etermination. Rather general versions of the Hary Littlewoo metho have been evelope in orer to investigate this problem, first by Davenport (see, in particular, [4, 5]), later by Birch [2], an most recently by Schmit [12], but unfortunately even Schmit s highly evelope version of the Hary Littlewoo metho is isappointingly ineffective in hanling systems of higher egree (see [11, 12]). In this paper we obtain explicit bouns for the number of variables require in Birch s Theorem by using a metho involving the Hary Littlewoo metho only inirectly, being base on a refinement of the elementary iagonalisation metho of Birch [1] first escribe in Wooley [15], where we restricte our investigations to systems of cubic an quintic forms. Although the size of our bouns may be aptly escribe as not even astronomical (an eloquent phrase of Birch), it seems that this paper contains the first truly explicit bouns in this problem. In orer to escribe our conclusions we require some notation. When k is a fiel, an r are natural numbers, an m is a non-negative integer, let,r (k) enote the least integer (if any such integer exists) with the property that whenever s >,r (k), an f i(x) k[x 1,..., x s ] (1 i r) are forms 1991 Mathematics Subject Classification: Primary 11D72, 11E76; Seconary 11G35, 14G05. The author is a Packar Fellow, an supporte in part by NSF grant DMS This paper was complete while the author was enjoying the hospitality of the Department of Mathematics at Princeton University. [79]

2 80 T. D. Wooley of egree, then the system of equations f i (x) = 0 (1 i r) possesses a solution set which contains a k-rational linear space of projective imension m. If no such integer exists, efine,r (k) to be +. We abbreviate v(0),r (k) to v,r (k), an efine φ,r (k) in like manner, save that the arbitrary forms of egree are restricte to be iagonal. When is even one has,r (Q) =, because efinite forms necessarily fail the real solubility conition, an thus we restrict attention to o. In previous work [15] we investigate systems of cubic an quintic forms. In particular, we showe that when r is a natural number an m is a nonnegative integer, then (1.1) 3,r (Q) < (90r)8 (log(27r)) 5 (m + 1) 5, an (1.2) 5,r (Q) < exp(1032 ((m + 1)r log(3r)) κ log(3r(m + 1))), where (1.3) κ = log 3430 log 4 = We note that earlier work of Lewis an Schulze-Pillot [8, inequality (4)] provies an estimate in many circumstances superior to (1.1), namely (1.4) 3,r (Q) r11 (m + 1) + r 3 (m + 1) 5. Moreover, if one seeks only the existence of a rational point on the intersection of r cubic hypersurfaces, then Schmit s boun v (0) 3,r (Q) < (10r)5 (see [10, Theorem 1]) is superior to both (1.1) an (1.4). We now exten our earlier conclusions to arbitrary systems of forms of o egree, an this will entail recoring some further notation. Suppose that A is a subset of R an Ψ is a function mapping A into A. When α is a real number, write [α] for the largest integer not exceeing α. Then we aopt the notation that whenever x an y are real numbers with x 1, then Ψ x (y) enotes the real number a [x], where (a n ) n=1 is the sequence efine by taking a 1 = Ψ(y), an a i+1 = Ψ(a i ) (i 1). Finally, when n is a nonnegative integer we efine the functions ψ (n) (x) by taking ψ (0) (x) = exp(x), an when n > 0 by putting (1.5) ψ (n) (x) = ψ (n 1) 42 log x (x). Theorem 1. Let be an o integer exceeing 5, an let r an m be non-negative integers with r 1. Then,r (Q) < ψ(( 5)/2) (r(m + 1)). The upper boun containe in Theorem 1 provies the first entirely explicit version of Birch s Theorem for systems of forms of equal egree, cor-

3 Birch s Theorem 81 responing conclusions for mixe egree systems following almost trivially (see Theorem 5.1 below). While systems of forms possessing mil singularities are susceptible to more powerful analyses (see Birch [2], or Tartakovskiĭ [13] for earlier but more restricte work), we stress that our aim in this paper is to provie generally applicable bouns free of any geometric hypotheses. We note that the constant 42 occurring in the efinition (1.5) can be reuce with greater effort, especially for large values of the parameters. However, it oes not appear feasible to aapt the methos of this paper to replace the level of recursion in (1.5) by any function appreciably smaller than log x. In this context we remark that even Birch s original metho [1] coul, with sufficient effort, be employe to yiel an explicit boun for,r (Q), though such a boun woul be extraorinarily weak by comparison with that provie by Theorem 1. An alternative to Birch s elementary metho for bouning,r (Q) is provie by Schmit s sophisticate version of the Hary Littlewoo metho, escribe briefly in [12]. We iscuss the quality of the bouns which may be wrought from such ieas in an appenix ( 6 of this paper). Although we o not carry out sufficient calculations to be confient of the precise bouns stemming from Schmit s methos, our analysis inicates that they yiel bouns qualitatively no stronger than (1.6) v (0),r (Q) < φ(( 5)/2) (r) ( 7), where φ (0) (x) = exp(x), an φ (n) (x) = φ (n 1) Ax (x) (n 1), for a suitable positive constant A. The superiority of Theorem 1 is plain. In this context we note that loose remarks concluing 2 of Schmit [12] might leave the impression that for a suitable function f, the methos of that section will establish a boun of the shape v (0),r (Q) < exp 3(f()r) ( 5). As shoul be clear from 6, however, a boun of the latter strength is wholly beyon reach of such methos when 7. Our proof of Theorem 1 is base on an inuctive strategy which epens for its success on an efficient iagonalisation process escribe in Wooley [15]. We begin, in 2, by recalling several of the key lemmata of [15] crucial to our subsequent arguments. Here an elsewhere in the proof of Theorem 1, we must extract conclusions simple enough that it remains feasible to keep control of the ensuing inuction, yet retain sufficient precision to preserve the quality of the ultimate bouns. It might be sai that the construction of a compromise between the latter two objectives represents the major ifficulty of our argument. In 3 we consier systems of septic forms, bouns on 7,r (Q) forming the basis for our main inuction. The main inuctive

4 82 T. D. Wooley step itself, in which we boun,r (Q) in terms of v(m) 2,R (Q) for suitable M an R, is establishe in 4. The proof of Theorem 1 is then complete routinely in 5 by making use of the main conclusions of 3 an 4. Finally, in 6, we provie an appenix in which we iscuss Schmit s metho an its consequences for Birch s Theorem. Throughout, implicit constants in Vinograov s notation an epen at most on the quantities occurring as subscripts to the notation. 2. Preliminary lemmata: reuction to iagonal forms. In this section we recall the reuction formulae from Wooley [15] which relate the solubility of arbitrary systems of homogeneous polynomials to iagonal ones. In orer to escribe these formulae we require some aitional notation. Given an r-tuple of polynomials F = (F 1,..., F r ) with coefficients in a fiel k, enote by ν(f) the number of variables appearing explicitly in F. We are intereste in the existence of solutions, over k, of systems of homogeneous polynomial equations with coefficients in k. When such a solution set contains a linear subspace of the ambient space, we efine its imension to be that when consiere as a projective space. When is a positive o integer, enote by G (m) (r, r 2,..., r 1 ; k) the set of (r +r r 1 )-tuples of homogeneous polynomials, of which r i have egree i for i = 1, 3,...,, with coefficients in k, an which possess no non-trivial linear space of solutions of imension m over k. We efine D (m) (r, r 2,..., r 1 ; k) to be the corresponing set of iagonal homogeneous polynomials. We then efine (r) = (r, r 2,..., r 1 ; k) by an we efine φ (m) (r, r 2,..., r 1 ; k) = sup φ (m) g G (m) (r,r 2,...,r 1 ;k) (r) = φ (m) (r, r 2,..., r 1 ; k) by (r, r 2,..., r 1 ; k) = sup f D (m) (r,r 2,...,r 1 ;k) ν(g), ν(f). We observe for future reference that both (r) an φ (m) (r) are increasing functions of the arguments m an r i (i = 1, 3,..., ). For the sake of convenience we abbreviate (r, 0,..., 0; k) to,r (k), an note that w (0) (r, 0,..., 0; k) = v,r(k). We also abbreviate φ (m) (r, 0,..., 0; k) to φ (m),r (k), an write φ,r(k) for φ (0),r (k). Next, when m 2, we efine H (m) (r; k) to be the set of r-tuples, (F 1,..., F r ), of homogeneous polynomials of egree, with coefficients in k, for which no linearly inepenent k-rational vectors e 1,..., e m exist such that F i (t 1 e t m e m ) is a iagonal form in t 1,..., t m for 1 i r. We

5 then efine (r) = (r; k) by Birch s Theorem 83 (r; k) = sup h H (m) (r;k) ν(h). Further, we aopt the convention that w (1) (r; k) = 0. Note that w(m) (r; k) is an increasing function of the arguments m an r. Moreover, when s > (r; k) an F 1,..., F r are homogeneous polynomials of egree with coefficients in k possessing s variables, then there exist k-rational vectors e 1,..., e m with the property that F i (t 1 e t m e m ) is a iagonal form in t 1,..., t m for 1 i r. The efficient iagonalisation process of [15] allue to in the introuction is emboie in the following lemma, which is nothing other than Lemma 2.1 of [15]. Lemma 2.1. Let be an o integer with 3, an let r, n an m be natural numbers. Then where M = w (n) w (n+m) (r; k) s + w (M) 2 (R; k), (r; k), s = 1 + w(n) 2 (S; k), N an for 0 u ( 1)/2, ( ) s + 2u 2 R 2u+1 = r 2u 1 an = w(m) (r; k), ( ) n + 2u 2 S 2u+1 = r. 2u 1 We must still boun (r; k) in terms of w (M) (R; k), for suitable M an R, an this we choose to o simply in the following two lemmata. Lemma 2.2. Let be an o positive number, let r be a natural number, an let m be a non-negative integer. Then,r where M = (m + 1)(φ,r (k) + 1). P r o o f. This is Lemma 2.2 of [15]. (k) w(m) (r; k), Lemma 2.3. Let be an o positive number with 3, an let r 1, r 3,......, r be non-negative integers with r > 0. Then for each non-negative integer m one has where M =,r (k). (r, r 2,..., r 1 ; k) w (M) 2 (r 2,..., r 1 ; k), P r o o f. This is Lemma 2.3 of [15].

6 84 T. D. Wooley In orer to make use of Lemma 2.2 in our argument, we require an estimate for φ,r (Q). For this purpose we use the corollary to Theorem 1 of Brüern an Cook [3] emboie in the following lemma (this is Lemma 3.2 of [15]; see also Low, Pitman an Wolff [9]). We remark that earlier work of Davenport an Lewis [6], though weaker, woul suffice to provie a version of Theorem 1 only slightly inferior to that given. Lemma 2.4. Let an r be natural numbers with o. Then φ,r (Q) r 3 log(3r 2 ). Before embarking on the analysis of systems of septic forms containe in the next section, we erive some simple estimates for (r; Q) when = 3, 5. Although these estimates are significantly weaker than the best attainable, we emphasise again that our aim is to control the complexity of our subsequent machinations through the use of simple bouns. Since in this an future iscussions we will be working only in the rational fiel Q, we henceforth omit explicit mention of the unerlying fiel from our various notations. Lemma 2.5. Suppose that r 3, r 1 an m are non-negative integers with r 1 < 3r 2 3. Then 3 (r 3, r 1 ) < (3r 3 (m + 1)) 25. P r o o f. It follows from the argument of Lewis an Schulze-Pillot [8, inequality (4)] (see [15, Lemma 3.3]) that 3,r < (11r)11 (m + 1) + 50r 3 (m + 1) 5. Then whenever r 1 < 3r 2 3, one fins by elimination of implicit linear equations that 3 (r 3, r 1 ) = r 1 + 3,r 3 < 3r (11r 3 ) 11 (m + 1) + 50r 3 3(m + 1) 5. A moest calculation thus reveals that 3 (r 3, r 1 ) < ( )r 11 3 (m + 1) 5 < (3r 3 (m + 1)) 25, whence the lemma follows. Lemma 2.6. Suppose that r 5, r 3, r 1 an m are non-negative integers with r 1 3r 2 3 an r 3 < 3r 2 5. Then 5 (r 5, r 3, r 1 ) < exp((5r 5 (m + 1)) 49 ). P r o o f. Write v = 5,r 5. Then by Theorem 2 of [15] (see (1.2) an (1.3) above), one has v < exp(10 32 ((m + 1)r 5 log(3r 5 )) κ log(3r 5 (m + 1))),

7 Birch s Theorem 85 where κ = (log 3430)/(log 4). But Lemma 2.3 shows that 5 (r 5, r 3, r 1 ) w (v) 3 (r 3, r 1 ), whence, in view of the hypotheses of the statement of the lemma, an making use of Lemma 2.5, one fins that log 5 (r 5, r 3, r 1 ) < 25 log(9r 2 5) + 25( )((m + 1)r 5 log(3r 5 )) κ log(3r 5 (m + 1)). Thus, following a moicum of computation, we obtain the upper boun log 5 (r 5, r 3, r 1 ) < 25(log 9)r ( )((log 3)(m + 1)r 2 5) 7 < (5r 5 (m + 1)) 49, an this completes the proof of the lemma. 3. The basis for the inuction: systems of septic forms. We establish the bouns for,r recore in Theorem 1 by eriving corresponing bouns for (r), an these we establish by inuction on, bouning (r) in terms of w (M) 2 (R) for suitable M an R. Although it is possible to use the bouns for 5 (r) escribe in Lemma 2.6 as the basis for this inuction, such a strategy entails complications best avoie. Instea, in this section, we establish bouns for 7 (r) which later form the basis of our inuction. We begin with the iagonalisation process. Lemma 3.1. Suppose that m an r are natural numbers. Then 7 (r) < exp 5 log(7m) (7rm). P r o o f. When m an r are natural numbers, write (3.1) 7 (r) = exp 5 log(7m) (7rm). We aim to show that for each R an M one has (3.2) w (M) 7 (R) < w (M) 7 (R), whence the conclusion of the lemma follows. Note that by efinition, for each natural number R one has w (1) 7 (R) = 0, so that (3.2) certainly hols when M = 1. Next suppose that m > 1, an that for each R the inequality (3.2) hols whenever M < m. We will establish that (3.2) hols for each R when M = m, an thus (3.2) will follow for all R an M by inuction. Let m an r be natural numbers with m 2. Write n = [(m + 1)/2], an note that n < m. By Lemma 2.1 one has (3.3) 7 (r) w (2n) 7 (r) s + w (N) 5 (R),

8 86 T. D. Wooley where (3.4) N = w (n) 7 (r), s = 1 + w(n) 5 (S), an for 0 u 3, ( ) s + 5 2u R 2u+1 = r 6 2u an ( ) n + 5 2u S 2u+1 = r. 6 2u We first boun s. Write N = [w (n) 7 (r)], an note that since n < m, the inuctive hypothesis shows that N N. Note also that for 0 u 3 one has S 2u+1 rn 6 2u. Then the hypotheses require for the application of Lemma 2.6 to boun w (N) 5 (S) are satisfie, an we may conclue from (3.4) that s 1 + w (N) 5 (rn 2, rn 4, rn 6 ) w (N) 5 (rn 2, rn 4, 2rn 6 ), whence (3.5) s < exp((5rn 2 (N + 1)) 49 ). But by (3.1) one has (3.6) N w (n) 7 (r) = exp 5 log(7n)(7rn) an (3.7) N = [exp 5 log(7n) (7rn)] 7rn. Also, plainly, for each m 2 it follows from (3.1) that N exp 5 (7). Then by combining (3.5) an (3.7), we obtain (3.8) s exp((n + 1) 147 ) < exp(n 148 ). Finally, we boun 7 (r) by substituting (3.8) into (3.3). Note that for 0 u 3 one has R 2u+1 rs 6 2u. Then the hypotheses require for the application of Lemma 2.6 to boun w (N) 5 (R) are satisfie, an we may conclue that whence 7 (r) s + w (N) 5 (rs 2, rs 4, rs 6 ) w (N) 5 (rs 2, rs 4, 2rs 6 ), 7 (r) < exp((5rs 2 (N + 1)) 49 ). Write s = exp(n 148 ). Then by (3.7) an (3.8), one has 7 (r) < exp((s(n + 1)) 98 ) exp(s 196 ) We therefore euce from (3.6) that = exp 2 (196N 148 ) < exp 2 (N 149 ). 7 (r) < exp 3 (149 log N) < exp 3 (exp 5 log(7n) 1 (7rm)).

9 Birch s Theorem 87 But on noting that whenever m 2 one has ( [ ]) m log 7 5 log(7m) 2, 2 we may conclue from (3.1) that 7 (r) < exp 5 log(7m) (7rm) = 7 (r), thereby establishing the inequality (3.2) with M = m an R = r. In view of the comments concluing the first paragraph of the proof, this completes the proof of the lemma. A boun for 7,r may now be obtaine by inserting the conclusion of Lemma 3.1 into Lemma 2.2. Lemma 3.2. Suppose that m an r are non-negative integers with r 1. Then 7,r < exp 37 log(7r(m+1))(7r(m + 1)). P r o o f. By combining Lemmata 2.2 an 2.4 with Lemma 3.1, one obtains (3.9) 7,r < exp 5 log(7m)(7rm), where (3.10) M = 16464r(m + 1) log(147r) 16464(log 147)r 2 (m + 1) < 7 6 r 2 (m + 1). But it follows from (3.10) that an log(7m) < log(7 7 r 2 (m + 1)) 7 log(7r(m + 1)), log(7rm) < log(7 7 r 3 (m + 1)) 7 log(7r(m + 1)) exp(7r(m + 1)), an hence (3.9) provies the estimate 7,r < exp 35 log(7r(m+1))+2(7r(m + 1)). The conclusion of the lemma follows immeiately. Note that the conclusion of Lemma 3.2 establishes Theorem 1 when = 7. In orer to establish our main inuctive step, however, we require a slightly more general result. Lemma 3.3. Suppose that r 2u+1 (0 u 3) an m are non-negative integers with r 1 3r 2 3, r 3 3r 2 5 an r 5 < 3r 2 7. Then 7 (r) < exp 39 log(7r7 (m+1))(7r 7 (m + 1)).

10 88 T. D. Wooley P r o o f. By Lemma 2.3 one has 7 (r) w (v) 5 (r 5, r 3, r 1 ), where v = 7,r 7. But in view of the hypotheses concerning r 1, r 3, r 5, we may apply Lemma 2.6 together with Lemma 3.2 to conclue that log 2 7 (r) < 49 log(15r 2 7(1 + exp 37 log(7r7 (m+1))(7r 7 (m + 1)))) < 98 exp 37 log(7r7 (m+1)) 1(7r 7 (m + 1)) < exp 37 log(7r7 (m+1))(7r 7 (m + 1)). The esire conclusion is almost immeiate from the latter inequality. 4. The inuctive step: systems of forms of higher egree. Thus far we have establishe Theorem 1 for = 7. We next establish the inuctive step which permits us to prove Theorem 1 for larger exponents, our argument following in spirit the trail lai own in 3. Our argument will be much simplifie by making use of the following efinition. Definition. We say that the function Ψ : [1, ) [1, ) satisfies the exponential growth conition if it has erivatives of all orers on [1, ), an moreover for each non-negative integer n, one has for each x [1, ) that n Ψ(x) x n e x. When D is an o integer exceeing 5, we make use of the following hypothesis. Hypothesis H D (Ψ). For all natural numbers M, an all 1 2 (D+1)-tuples R = (R D, R D 2,..., R 1 ) of non-negative integers satisfying R D 2 < 3RD 2 an R i 3Ri+2 2 (i = 1, 3,..., D 4), one has (4.1) w (M) D (R) < Ψ(DR D(M + 1)). Our initial avance on the inuctive step is provie by the iagonalisation process. Lemma 4.1. Let be an o integer exceeing 7. Suppose that Ψ is a function satisfying the exponential growth conition, an suppose further that the hypothesis H 2 (Ψ) hols. Then whenever m an r are natural numbers, one has (r) < Ψ 5 log(m) (rm). P r o o f. When m an r are natural numbers, write (4.2) (r) = Ψ 5 log(m) (rm).

11 We aim to show that for each R an M one has (4.3) w (M) Birch s Theorem 89 (R) < w (M) (R), whence the conclusion of the lemma follows. Since for each natural number R one has w (1) (R) = 0, the inequality (4.3) certainly hols for M = 1. Suppose next that m > 1, an that for each R the boun (4.3) hols whenever M < m. We establish that (4.3) hols for each R when M = m, an so (4.3) hols for all R an M by inuction. Let m an r be natural numbers with m 2. Write n = [(m + 1)/2], an note that n < m. By Lemma 2.1 one has (4.4) where (r) w (2n) (r) s + w (N) 2 (R), (4.5) N = w (n) (r), s = 1 + w(n) 2 (S), an for 0 u ( 1)/2, ( ) s + 2u 2 R 2u+1 = r 2u 1 an ( ) n + 2u 2 S 2u+1 = r. 2u 1 Write N = [w (n) (r)], an note that since n < m, the inuctive hypothesis of this lemma shows that N N. Note also that for 0 u ( 1)/2 one has S 2u+1 rn 2u 1, so that the hypotheses require for the application of the hypothesis H 2 (Ψ) to boun w (N) 2 (S) are satisfie. We may therefore conclue from (4.5) that s 1 + w (N) 2 (rn2, rn 4,..., rn 1 ) w (N) 2 (rn2, rn 4,..., rn 3, 2rn 1 ), whence (4.6) s < Ψ(rn 2 (N + 1)). On recalling that Ψ satisfies the exponential growth conition, it follows from (4.2) that (4.7) N = [Ψ 5 log(n) (rn)] rn. Also, plainly, for each m 2 it follows from (4.2) that N exp 5 (). Then by combining (4.6) an (4.7) we euce that (4.8) s < Ψ((N + 1) 3 ) < Ψ(N 4 ). Having successfully boune s, we next estimate (r) by substituting (4.8) into (4.4). Note first that for 0 u ( 1)/2 one has R 2u+1 rs 2u 1, so that the hypotheses require for the application of the hypothesis H 2 (Ψ) to boun w (N) 2 (R) are satisfie. We therefore euce

12 90 T. D. Wooley from (4.4) that whence (r) s + w (N) 2 (rs2, rs 4,..., rs 1 ) w (N) 2 (rs2, rs 4,..., rs 3, 2rs 1 ), (r) < Ψ(rs 2 (N + 1)). Write s = Ψ(N 4 ). Then on recalling that Ψ satisfies the exponential growth conition, it follows from (4.7) an (4.8) that (r) < Ψ((s(N + 1)) 2 ) Ψ(s 4 ) = Ψ((Ψ(N 4 )) 4 ) Ψ 2 (4N 4 ) < Ψ 2 (N 5 ). On making use of the boun N w (n) (r) = Ψ 5 log(n)(rn), therefore, we conclue that (r) < Ψ 3 (5Ψ 1 (N)) Ψ 3 (Ψ 5 log(n) 1 (rm)). But on noting that whenever m 2 one has ( [ ]) m log 5 log(m) 2, 2 we fin from (4.2) that (r) < Ψ 5 log(m) (rm) = (r), so that (4.3) hols with M = m an R = r. On recalling the comments concluing the first paragraph of the proof, the lemma now follows by inuction. Following the lea provie in 3, we next boun,r on the hypothesis H 2 (Ψ) by combining the conclusions of Lemmata 4.1 an 2.2. Lemma 4.2. Let be an o integer exceeing 7. Suppose that Ψ is a function satisfying the exponential growth conition, an suppose further that the hypothesis H 2 (Ψ) hols. Then whenever m an r are non-negative integers with r 1, one has,r < Ψ 41 log(r(m+1))(r(m + 1)). P r o o f. On combining Lemmata 2.2 an 2.4 with Lemma 4.1, one obtains (4.9),r where (4.10) < Ψ 5 log(m)(rm), M = 48(m + 1)r 3 log(3r 2 ) 48(log 3) 5 r 2 (m + 1) < 7 r 2 (m + 1).

13 It follows from (4.10) that Birch s Theorem 91 log(m) < 8 log(r(m + 1)) an log(rm) < exp(r(m + 1)), an hence (4.9) leas to the upper boun,r < Ψ 40 log(r(m+1))+2(r(m + 1)). The conclusion of the lemma follows immeiately. In orer to complete the inuctive step we must combine the conclusion of the latter lemma with the hypothesis H 2 (Ψ) in orer to boun (r). Lemma 4.3. Let be an o integer exceeing 7. Suppose that Ψ is a function satisfying the exponential growth conition, an suppose further that the hypothesis H 2 (Ψ) hols. Then whenever r 2u+1 (0 u 1 2 ( 1)) an m are non-negative integers with r i 3ri+2 2 (i = 1, 3,..., 4) an r 2 < 3r 2, one has (r) < Ψ 42 log(r (m+1))(r (m + 1)). P r o o f. By Lemma 2.3 one has (r) w (v) 2 (r 2,..., r 1 ), where v =,r. The hypotheses concerning r i for i = 1, 3,..., 2 permit us the use of the hypothesis H 2 (Ψ) to boun w (v) 2 (r 2,..., r 1 ), an thus on employing Lemma 4.2 to boun v, we euce that w (v) 2 (r 2,..., r 1 ) < Ψ(3( 2)r 2 (v + 1)) < Ψ 2 (2Ψ 41 log(r (m+1)) 1(r (m + 1))) Ψ 41 log(r (m+1))+2(r (m + 1)). The conclusion of the lemma is now immeiate. 5. The proof of Theorem 1. The machinery of our argument now fully assemble, we crank up the inuction which establishes Theorem 1. Note first that, in view of Lemma 3.3, the hypothesis H 7 (ψ (1) ) hols, where ψ (1) is efine by (1.5). Moreover, ψ (1) plainly satisfies the exponential growth conition. Suppose next that is an o integer exceeing 7, an that the hypothesis H 2 (ψ (( 7)/2) ) hols. Since, plainly, ψ (( 7)/2) also satisfies the exponential growth conition, it follows from Lemma 4.3 that the hypothesis H (ψ (( 5)/2) ) hols. We therefore euce, by inuction, that the hypothesis H (ψ (( 5)/2) ) hols for every o integer exceeing 5. Consequently, on applying Lemma 4.2, we conclue that the inequality,r < ψ(( 7)/2) 41 log(r(m+1)) (r(m + 1)) < ψ(( 5)/2) (r(m + 1))

14 92 T. D. Wooley hols for every o integer exceeing 5. This completes the proof of Theorem 1. We conclue this section by proviing a boun, similar to that recore in Theorem 1, applicable to systems of forms of mixe egrees. Theorem 5.1. Let be an o integer exceeing 5, an let r 1, r 3,..., r an m be non-negative integers with r 1. Then (r,..., r 1 ; Q) < ψ (( 5)/2) ((r 1 + r r )(m + 1)). P r o o f. Let s be any integer exceeing ψ (( 5)/2) ((r 1 + r r ) (m + 1)), an let F ij (x) Z[x 1,..., x s ] (1 j r i ) be homogeneous polynomials of egree i for i = 1, 3,...,. We aim to show that the system (5.1) F ij (x) = 0 (1 j r i ; i = 1, 3,..., ) possesses a solution x Z s \ {0}, whence the theorem follows. We efine a new system of equations by writing (5.2) G ij (x) = (x x x 2 s) ( i)/2 F ij (x) (1 j r i ; i = 1, 3,..., ). Plainly, each polynomial G ij (x) is homogeneous of egree, an thus the system of equations (5.3) G ij (x) = 0 (1 j r i ; i = 1, 3,..., ) consists of r 1 + r r equations of egree. In view of the efinition of s, it follows from Theorem 1 that the system (5.3) necessarily possesses a linear space of rational solutions of projective imension m, whence by (5.2) the system (5.1) also possesses such a solution. This completes the proof of the theorem. 6. Appenix: Bouns stemming from Schmit s metho. Our iscussion of upper bouns for,r woul be incomplete without mention of Schmit s sophisticate version of the circle metho, which itself provies an interesting approach to bouning v (0),r, for o exponents. Before outlining the strategy escribe by Schmit in 2 of [12], we require some notation. When F Q[x] is a form of egree > 1, write h(f ) for the least number h such that F may be written in the form h F = A i B i, i=1 with A i, B i forms in Q[x] of positive egree. When F 1,..., F r are forms of equal egree > 1, an F = (F 1,..., F r ), write h(f) = min h(f ), where

15 Birch s Theorem 93 the minimum is taken over forms F lying in the rational pencil of F, which is to say the set {F Q[x] : F = c 1 F c r F r an (c 1,..., c r ) Q r \ {0}}. The important achievement of Schmit in [12] is the evelopment of a version of the Hary Littlewoo metho which, given a system of forms F Q[x] r with h(f) large enough, establishes an asymptotic formula of the expecte shape for the number of integral zeros of F insie a large box. Here, formula of the expecte shape simply means the prouct of local ensities, emboie in the prouct of the singular series an singular integral familiar to experts in the circle metho. As Schmit [12, 2] observes, such a result offers the possibility of a reuction process which, given sufficiently many variables, establishes the existence of a non-trivial rational zero to a given system. Roughly speaking, one observes that for forms of o egree, whenever h(f) is large enough Schmit s circle metho alreay establishes the existence of a non-trivial rational zero. On the other han, if h(f) is not large enough, say h(f) h 0, then some form F lying in the rational pencil of F ecomposes in the shape h 0 F = A i B i, i=1 with A i, B i forms in Q[x] of positive egree. Since the forms F i are of o egree, one may assume without loss of generality that the A i are all of o egree. Consequently, the system F is soluble non-trivially provie that there is a non-trivial solution to the system F(x) = 0 an A i (x) = 0 (1 i h 0 ), where F enotes the system F with a suitable form elete. Since the A i all have o egree smaller than the elete form, one perceives an obvious reuctive strategy which ultimately either shows the system to be non-trivially soluble, or else reuces it to a system of linear equations, which again is non-trivially soluble. We summarise this approach in quantitative form in the following lemma. Lemma 6.1. Let be an o integer exceeing 1, let r 1, r 3,..., r be non-negative integers with r 1, an efine Then (6.1) w (0) (r; Q) max h (r; k) = 2 4k k!r k w (0) (r; Q p) (3 k ). k=3,5,..., w(0) (r, r 2,..., r k 1, r k 2 + h (r; k), r k 4,..., r 1 ; Q).

16 94 T. D. Wooley P r o o f. The conclusion (6.1) is immeiate from Theorem II of [12] an its supplement, on applying the argument escribe by Schmit in 2 of [12]. We are now in a position to illustrate the use of Schmit s strategy. We concentrate on systems of septic forms, such systems isplaying the salient features an problems of the metho. Although we will be somewhat rough in our estimates, it shoul be clear that Lemma 6.1 is incapable of oing substantially better. Consier bouns for w (0) 7 (r 7, r 5, r 3, 0). Write R = R(r) = r 7 +r 5 +r 3, an suppose throughout that R is large. We recall an immeiate consequence of Corollary 1.1 of Wooley [14] (improving on earlier work of Leep an Schmit [7]), which provies the estimate w (0) (r, r 2,..., r 1 ; Q p ) (r + r r 1 ) It follows that there is an integral constant c 1 > 0 such that whenever r 5 > 0, the best bouns stemming from Lemma 6.1 cannot be substantially stronger than (6.2) w (0) 7 (r 7, r 5, r 3, 0) w (0) 7 (r 7, r 5 1, r 3 + R c 1, 0). Meanwhile, when r 5 = 0 an r 7 > 0, there is an integral constant c 2 > 0 such that the corresponing best available boun from Lemma 6.1 cannot be substantially stronger than (6.3) w (0) 7 (r 7, 0, r 3, 0) w (0) 7 (r 7 1, R c 2, r 3, 0). We next boun v (0) 7,r by use of (6.2) an (6.3), as suggeste by Schmit s strategy. When r is large, one obtains v (0) 7,r = w(0) 7 (r, 0, 0, 0) w(0) 7 (r 1, rc 2, 0, 0) w (0) 7 (r 1, rc 2 1, c 3 r c 1c 2, 0), for an integral constant c 3 > 0. Next, by repeate application of (6.2) one obtains for n = 1,..., r c 2 the estimate w (0) 7 (r, 0, 0, 0) w(0) 7 (r 1, rc 2 n, c n+2 r cn 1 c 2, 0), with c n > 0 uniformly boune above by a fixe real number. Consequently, when r is large enough one has v (0) 7,r w(0) 7 (r 1, 0, [exp 2(r a 1 )], 0), for a suitable fixe real number a 1 > 0. We next apply (6.3), an then again repeately apply (6.2). We now obtain v (0) 7,r w(0) 7 (r 2, 0, [exp 4(r a 2 )], 0),

17 Birch s Theorem 95 for a suitable fixe real number a 2 > 0. Repeating this iteration pattern, we ultimately obtain v (0) 7,r w(0) 7 (0, 0, [exp 2r(r a )], 0), for a suitable fixe real number a > 0. It therefore follows from Schmit [10, Theorem 1] that (6.4) v (0) 7,r exp 2r+1(r). Notice that in eriving the upper boun (6.4), it is conceivable that in following Schmit s strategy, we are force by the structure of our implicit forms to apply the bouns (6.2) an (6.3) in the inicate fashion. Thus we conclue that Schmit s metho is incapable of proviing estimates substantially stronger than (6.4). By contrast, Theorem 1 of the present paper establishes the boun v (0) 7,r exp 42 log(7r)(7r), which is plainly stronger for large r. For larger o exponents, the worst bouns that arise from Schmit s strategy are obtaine as follows. One repeately applies Lemma 6.1 so as to reuce the number of implicit equations of lowest egree until none of that egree remain. Here one ignores implicit linear equations. One then takes the next lowest egree for which there are implicit equations, applies Lemma 6.1 so as to reuce the number of such equations, this in turn spawning many equations of lower egree. An analysis only slightly more careful than that above will reveal that the bouns attainable by such an approach take the shape (1.6) inicate in the introuction. References [1] B. J. B i r c h, Homogeneous forms of o egree in a large number of variables, Mathematika 4 (1957), [2], Forms in many variables, Proc. Roy. Soc. Lonon Ser. A 265 (1961/62), [3] J. Brüern an R. J. Cook, On simultaneous iagonal equations an inequalities, Acta Arith. 62 (1992), [4] H. Davenport, Cubic forms in thirty-two variables, Philos. Trans. Roy. Soc. Lonon Ser. A 251 (1959), [5], Cubic forms in 16 variables, Proc. Roy. Soc. Lonon Ser. A 272 (1963), [6] H. Davenport an D. J. Lewis, Simultaneous equations of aitive type, Philos. Trans. Roy. Soc. Lonon Ser. A 264 (1969), [7] D. B. L e e p an W. M. S c h m i t, Systems of homogeneous equations, Invent. Math. 71 (1983), [8] D. J. Lewis an R. Schulze-Pillot, Linear spaces on the intersection of cubic hypersurfaces, Monatsh. Math. 97 (1984), [9] L. Low, J. Pitman an A. Wolff, Simultaneous iagonal congruences, J. Number Theory 36 (1990), 1 11.

18 96 T. D. Wooley [10] W. M. S c h m i t, On cubic polynomials IV. Systems of rational equations, Monatsh. Math. 93 (1982), [11], Analytic methos for congruences, Diophantine equations an approximations, in: Proceeings of the International Congress of Mathematicians (Warsaw, 1983), Vol. 1, PWN, Warszawa, 1984, [12], The ensity of integer points on homogeneous varieties, Acta Math. 154 (1985), [13] V. A. Tartakovskiĭ, Die asymptotische Gesetze er allgemeinen Diophantischen Analyse mit vielen Unbekannten, Izv. Aka. Nauk SSSR Ser. Mat.-Fiz. 1935, [14] T. D. Wooley, On the local solubility of iophantine systems, Compositio Math. 111 (1998), [15], Forms in many variables, in: Analytic Number Theory: Proceeings of the 39th Taniguchi International Symposium, Kyoto, May 1996, Y. Motohashi (e.), Lonon Math. Soc. Lecture Note Ser. 247, Cambrige Univ. Press, 1997, Department of Mathematics University of Michigan East Hall, 525 East University Avenue Ann Arbor, Michigan U.S.A. Current aress: Department of Mathematics Princeton University Fine Hall, Washington Roa Princeton, New Jersey U.S.A. wooley@math.lsa.umich.eu wooley@math.princeton.eu Receive on (3270)