The dimension growth conjecture, polynomial in the degree and without logarithmic factors
Wouter Castryck, Raf Cluckers, Philip Dittmann, Kien Huu Nguyen
aa r X i v : . [ m a t h . N T ] A p r THE DIMENSION GROWTH CONJECTURE, POLYNOMIAL INTHE DEGREE AND WITHOUT LOGARITHMIC FACTORS
WOUTER CASTRYCK, RAF CLUCKERS, PHILIP DITTMANN, AND KIEN HUU NGUYEN
Abstract.
We address Heath-Brown’s and Serre’s dimension growth conjecture(proved by Salberger), when the degree d grows. Recall that Salberger’s dimensiongrowth results give bounds of the form O X,ε ( B dim X + ε ) for the number of rationalpoints of height at most B on any integral subvariety X of P n Q of degree d ≥ , where one can write O d,n,ε instead of O X,ε as soon as d ≥ . Our maincontribution is to remove the factor B ε as soon as d ≥ , without introducing afactor log B , while moreover obtaining polynomial dependence on d of the impliedconstant. Working polynomially in d allows us to give a self-contained and slightlysimplified treatment of dimension growth for degree d ≥ , while in the range ≤ d ≤ we invoke results by Browning, Heath-Brown and Salberger. Along theway we improve the well-known bounds due to Bombieri and Pila on the numberof integral points of bounded height on affine curves and those by Walsh on thenumber of rational points of bounded height on projective curves. The formerimprovement leads to a slight sharpening of a recent estimate due to Bhargava,Shankar, Taniguchi, Thorne, Tsimerman and Zhao on the size of the -torsionsubgroup of the class group of a degree d number field. Our treatment builds onrecent work by Salberger which brings in many primes in Heath-Brown’s variant ofthe determinant method, and on recent work by Walsh and Ellenberg–Venkatesh,who bring in the size of the defining polynomial. We also obtain lower boundsshowing that one cannot do better than polynomial dependence on d . Introduction and main results X of degree d , see [28, p. 27] and [27, p. 178], which was dubbed the dimensiongrowth conjecture by Browning in [5]. The question puts forward concrete upperbounds on the number of such points with height at most B , as a function of B . Mathematics Subject Classification.
Primary 11D45, 14G05; Secondary 11G35.The authors would like to thank Gal Binyamini, Jonathan Pila, Arne Smeets, Jan Tuitman,and Alex Wilkie for interesting discussions on the topics of the paper, and Per Salberger forsharing his preprint with us and for interesting exchanges of ideas. The authors W.C., R.C.,and K.H.N. were partially supported by the European Research Council under the EuropeanCommunity’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr.615722 MOTMELSUM and thank the Labex CEMPI (ANR-11-LABX-0007-01). The author W.C.is affiliated on a voluntary basis with the research group imec-COSIC at KU Leuven and with theDepartment of Mathematics: Algebra and Geometry at Ghent University. The authors R.C.and P.D. are partially supported by KU Leuven IF C14/17/083. The author K.H.N. is partiallysupported by the Fund for Scientific Research - Flanders (Belgium) (FWO) 12X3519N.
This dimension growth conjecture is now a theorem due to Salberger [26] (and othersunder various conditions on d ); moreover, for d ≥ Salberger obtains completeuniformity in X , keeping only d and the dimension of the ambient projective spacefixed, thereby confirming a variant that had been proposed by Heath-Brown.We remove from these bounds the factors of the form B ε when the degree d isat least , without creating a factor log B , while moreover obtaining polynomialdependence on d of the constants. The approach with polynomial dependence in d is implemented in all auxiliary results as well, and it in fact allows us to give a moredirect and self-contained proof of the dimension growth conjecture for d at least ; our treatment of the cases d = 5 , . . . , is not self-contained and uses [7] when d > and a result from [26] for d = 5 . In particular, we obtain similar improvementsto bounds of Bombieri–Pila on the number of integral points of bounded height onaffine irreducible curves [3, Theorem 5], and by Walsh on the number of rationalpoints of bounded height on integral projective varieties [30, Theorems 1.1, 1.2, 1.3].The possibility of polynomial dependence on d came to us via a question raisedby Yomdin (see below Remark 3.8 of [8]) in combination with the determinantmethod with smooth parameterizations as in [23], refined in [10], and via the workby Binyamini and Novikov [2, Theorem 6]. The removal of the factor B ε withoutneeding log B is recently achieved by Walsh [30, Theorems 1.1, 1.2, 1.3] who com-bines ideas by Ellenberg and Venkatesh [11] with the determinant method basedon p -adic approximation (rather than on smooth maps) due to Heath-Brown [17],refined in [26].We point out a difference between the dimension growth conjecture and the con-text of Manin’s conjecture [20]: the bounds in the former are valid for all heights B ≥ while for the latter, the asymptotics for large B are studied. For furthercontext we refer to [5].1.2. Let us make all this more precise. We study the number N ( X, B ) of rational points of height at most B on subvarieties X of P n defined over Q . Here,the height H ( x ) of a Q -rational point x in P n is given by H ( x ) = max( | x | , . . . , | x n | ) for an n + 1 -tuple ( x , . . . , x n ) of integers x i which are homogeneous coordinates for x and have greatest common divisor equal to .Salberger proves in [26] the so-called dimension growth conjecture raised as aquestion by Serre in [28, p. 27] following Heath-Brown’s question [16, p. 227] : Dimension Growth [26, Theorem 0.1] If X is an integral projectivevariety of degree d ≥ defined over Q , then N ( X, B ) ≤ O X,ε ( B dim X + ε ) . One should compare the bound for N ( X, B ) from this theorem to the trivial upperbound O d,n ( B dim X +1 ) that follows from Lemma 4.1.1 below. HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 3
A variant of this question from [27, p. 178] replaces the factor B ε by log( B ) c forsome c depending on X , see Example 1.4 below.Heath-Brown [17] introduces a form of this conjecture with uniformity in X forfixed d and n , and he develops a new variant of the determinant method using p -adicapproximation instead of smooth parameterizations as in [3], [2], [23], [10]. In [26],Salberger proves this uniform version of the dimension growth conjecture for d ≥ . Uniform Dimension Growth [26, Theorem 0.3] For X ⊆ P n Q anintegral projective variety of degree d ≥ , one has N ( X, B ) ≤ O d,n,ε ( B dim X + ε ) . Almost all situations of this uniform dimension growth had been obtained previ-ously in [17] and [7], including the case d = 2 but without the (difficult) cases d = 4 and d = 5 . Our main contributions are to make the dependence on d polynomial, toremove the factor B ε without having to use factors log B , and to provide relativelyself-contained proofs for large degree, with main result as follows.1. Theorem (Uniform dimension growth) . Given n > , there exist constants c = c ( n ) and e = e ( n ) , such that for all integral projective varieties X ⊆ P n Q of degree d ≥ and all B ≥ one has (1.2.1) N ( X, B ) ≤ cd e B dim X . In a way, one cannot do better than polynomial dependence on d , see the lowerbounds from Proposition 5 and Section 6 below.We heavily rework results and methods of Salberger, Walsh, Ellenberg–Venkatesh,Heath-Brown, and Browning, and use various explicit estimates for Hilbert func-tions, for certain universal Noether polynomials as in [24], and for solutions of linearsystems of equations over Z from [4].1.3. Rational points on curves and surfaces.
Let us make precise some of ourimprovements for counting points on curves and surfaces, which are key to Theorem1. We obtain the following improvement of Walsh’s Theorem 1.1 [30].2.
Theorem (Projective curves) . Given n > , there exists a constant c = c ( n ) such that for all d > and all integral projective curves X ⊆ P n Q of degree d and all B ≥ one has N ( X, B ) ≤ cd B /d . In view of Proposition 5 below, the exponent of d in Theorem 2 can perhaps belowered, but cannot become lower than in general. Several adaptations of resultsand proofs of [30] are key to our treatment and are developed in Section 3.For affine counting we use the following notation for a variety X ⊆ A n Q and apolynomial f in Z [ y , . . . , y n ] : N aff ( X, B ) := { x ∈ Z n | | x i | ≤ B for each i and x ∈ X ( Q ) } , and N aff ( f, B ) := { x ∈ Z n | | x i | ≤ B for each i and f ( x ) = 0 } . CASTRYCK, CLUCKERS, DITTMANN, AND NGUYEN
By a careful elaboration of the argument from [11, Remark 2.3] and an explicitbut otherwise classical projection argument, we find the following improvement ofbounds by Bombieri–Pila [3, Theorem 5] and later sharpenings by Pila [21], [22],Walkowiak [29], Ellenberg–Venkatesh [11, Remark 2.3], Binyamini and Novikov [2,Theorem 6], and others.3.
Theorem (Affine curves) . Given n > , there exists a constant c = c ( n ) suchthat for all d > , all integral affine curves X ⊆ A n Q of degree d , and all B ≥ onehas N aff ( X, B ) ≤ cd B /d (log B + d ) . A variant of Theorem 3 is given in Section 4, where log B is absent and insteadthe size of the coefficients of the polynomial f defining the affine planar curve comesin. It is well-known that Theorems 1, 2, 3 imply similar bounds for varieties definedand integral over Q (instead of Q ), by intersecting with a Galois conjugate andusing a trivial bound, see Lemma 4.1.3. The following improves Theorem 0.4 of [26]and is key to Theorem 1. It can be seen as an affine form of the dimension growththeorem, for hypersurfaces.4. Theorem (Affine hypersurfaces) . Given n > , there exist constants c = c ( n ) and e = e ( n ) , such that for all polynomials f in Z [ y , . . . , y n ] whose homogeneouspart of highest degree h ( f ) is irreducible over Q and whose degree d is at least , onehas N aff ( f, B ) ≤ cd e B n − . One should compare the bound from this theorem to the trivial upper bound O d,n ( B n − ) from Lemma 4.1.1.1.4. Example and a question.
In Serre’s example [27, p. 178] of the degree surface in P given by the equation xy = zw , the logarithmic factor log B cannotbe dispensed with in the upper bound. Hence, (1.2.1) of Theorem 1 cannot holdfor d = 2 in general. For d = 3 , the bound from (1.2.1) remains wide open sincealready uniformity in X ⊆ P n of degree is not known for the uniform dimensiongrowth with O d,n,ε ( B dim X + ε ) as upper bound (see [26] for subtleties when d = 3 ).For d = 4 , one may investigate whether (1.2.1) of Theorem 1 remains true, that is,without involving a factor B ε or log B .1.5. Lower bounds.
In Section 6 we discuss the necessity of the polynomial de-pendence on d in the above theorems.5. Proposition.
For each integer d > there is an integral projective curve X ⊆ P of degree d and an integer B ≥ such that d B /d ≤ N ( X, B ) . In particular, in the statement of Theorem 2 it is impossible to replace the factor d with an expression in d which is o ( d ) . HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 5
Similarly we show that it is impossible to replace the quartic dependence on d of the bound from Theorem 3 by a function in o ( d / log d ) . We also show that inTheorems 1 and 4 we cannot take e < resp. e < .1.6. An application.
When it comes to applications, our bounds can be usedas substitutes for those by Salberger, Bombieri–Pila, and Walsh upon which theyimprove, potentially leading to stronger statements. A very recent example of suchan application is Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao’sbound [1, Theorem 1.1] on the number h ( K ) of -torsion elements in the classgroup of a degree d > number field K , in terms of its discriminant ∆ K . Precisely,they show that h ( K ) ≤ O d,ε ( | ∆ K | − d + ε ) , thereby obtaining a power saving over the trivial bound coming from the Brauer-Siegel theorem. This power saving is mainly accounted for by an application ofBombieri and Pila’s bound from [3, Theorem 5]. In Section 4 we explain how ourimproved bound stated in Theorem 3, or rather its refinement stated in Corol-lary 4.2.4, allows for removal of the factor | ∆ K | ε as soon as d is odd; if d is eventhen we can replace it by log | ∆ K | .6. Theorem.
For all degree d > number fields K we have h ( K ) ≤ O d ( | ∆ K | − d (log | ∆ K | ) − ( d mod 2) ) . It is possible to make the hidden constant explicit, but targeting polynomialgrowth seems of lesser interest since | ∆ K | is itself bounded from below by an expo-nential expression in d , coming from Minkowski’s bound.1.7. Structure of the paper.
In Section 2 we render several results of Salberger[25] explicit in terms of the degrees and dimensions involved. In Section 3 wesimilarly adapt the results of Walsh [30]. Section 4 completes the proofs of ourmain results in the hypersurface case, which is complemented by Section 5, in whichwe discuss projection arguments from [7], explicit in the degrees and dimensions,and thus finish the proofs of our main theorems. Finally, in Section 6, we providelower bounds showing the necessity of polynomial dependence on d in our mainresults. 2. The determinant method for hypersurfaces
With the aim of improving the results of [30] in the next section, we sharpensome results from Salberger’s global determinant method. The main result of thissection is Corollary 2.9, which improves on [26, Lemmas 1.4, 1.5] (see also [30,Theorem 2.2]). This mainly depends on making [25, Main Lemma 2.5] in the caseof hypersurfaces explicit in its independence of the degree.Let f be an absolutely irreducible homogeneous polynomial in Z [ x , . . . , x n +1 ] which is primitive, and let X be the hypersurface in P n +1 Q defined by f . For p aprime number, let X p denote the reduction of X modulo p , i.e. the hypersurface in P n +1 F p described by the reduction of f mod p . CASTRYCK, CLUCKERS, DITTMANN, AND NGUYEN
Lemma (Lemma 2.3 of [25], explicit for hypersurfaces) . Let A be the stalk of X p at some F p -point P of multiplicity µ and let m be the maximal ideal of A . Let g X,P : Z > → Z be the function given by g X,P ( k ) = dim A/ m m k / m k +1 for k > . Thenone has g ( k ) = (cid:18) n + kn (cid:19) for k < µ and g ( k ) = (cid:18) n + kn (cid:19) − (cid:18) n + k − µn (cid:19) for k ≥ µ. In particular, g ( k ) ≤ µk n − ( n − O n ( k n − ) for all k ≥ , where the implied constant depends only on n , as indicated.Proof. The function g is identical to the Hilbert function of the projectivized tangentcone of X p at P , which is a degree µ hypersurface in P n . This gives the explicitexpression for g , so it only remains to prove the estimate.Consider first k < µ . Then g ( k ) = (cid:18) n + kn (cid:19) = k n n ! + ( n + 1) k n − n − O n ( k n − ) . Since µ > k , for k ≥ n we immediately obtain the desired inequality, and the k between and n are covered by choosing the constant large enough.Now consider k ≥ µ . Write p ( X ) for the polynomial (cid:0) n + Xn (cid:1) and a i for its coeffi-cients. Then p ( k ) − p ( k − µ ) = a n ( k n − ( k − µ ) n ) + a n − ( k n − − ( k − µ ) n − ) + O n ( k n − ) Observe that a n = 1 /n ! , a n − = ( n + 1) / (2( n − a n ( n + 1) n/ , and write k n − ( k − µ ) n = µ ( k n − + ( k − µ ) k n − + · · · + ( k − µ ) n − ) as well as k n − − ( k − µ ) n − = µ ( k n − + · · · + ( k − µ ) n − ) . Considering µ ≥ n ( n + 1) / , we have ( k − µ ) i k n − − i + ( n + 1) n k − µ ) i − k n − − i ≤ k n − for i ≥ , and hence a n ( k n − ( k − µ ) n ) + a n − ( k n − − ( k − µ ) n − ) ≤ µ/n !( k n − + · · · k n − ) = µk n − / ( n − as desired. The finitely many µ less than n ( n + 1) / are again taken care of bytaking a sufficiently large constant in the O n . (cid:3) HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 7
Lemma.
Let c, n, µ > be integers. Let g : Z ≥ → Z ≥ be a function with g (0) = 1 and satisfying g ( k ) ≤ µk n − ( n − + ck n − for k > . Let ( n i ) i ≥ be the non-decreasing sequence of integers m ≥ where m occurs exactly g ( m ) times. Then forany s ≥ we have n + · · · + n s ≥ ( n ! µ ) /n nn + 1 s /n − O n,c ( s ) . This statement is implicitly contained in the proof of [25, Main Lemma 2.5], butwe give the full proof to stress that the error term does not depend on µ . Proof.
Note that replacing g by a function which is pointwise larger than g at anypoint only strengthens the claim, so we may as well assume that g ( k ) = µn ! ( k n − ( k − n ) + c ( k n − − ( k − n − ) for k > . Let G : Z ≥ → Z ≥ be given by G ( k ) = g (0)+ · · · + g ( k ) = µn ! k n + ck n − +1 .Now ( n ! µ ) /n nn + 1 G ( k ) /n = µk n +1 ( n − n + 1) + O n,c ( k n ) , and g (0)+ · · · + kg ( k ) ≥ µ ( n − X i ≤ k ( i n + O n ( ci n − )) = µ ( n − n + 1) k n +1 + O n,c ( µk n ) . This proves the lemma for s = G ( k ) .To deduce the result for general s > , let k be the unique integer with G ( k −
Consider A as in Lemma 2.1, and let ( n i ( A )) i ≥ be the non-decreasingsequence of integers m ≥ where m occurs exactly dim A/ m m k / m k +1 times. Write A ( s ) = n ( A ) + · · · + n s ( A ) . Then A ( s ) ≥ ( n ! /µ ) /n ( n/ ( n + 1)) s /n − O n ( s ) , where the implied constant only depends on n .Proof. This is immediate from the last two lemmas. (cid:3)
As usual, write Z ( p ) for the localization of Z at (the complement of) the primeideal ( p ) .2.4. Lemma (Lemma 2.4 of [25], cited as in Appendix of [7]) . Let R be a noe-therian local ring containing Z ( p ) , A = R/pR , and consider ring homomorphisms ψ , . . . , ψ s : R → Z ( p ) . Let r , . . . , r s be elements of R . Then the determinant of the s × s -matrix ( ψ i ( r j )) is divisible by p A ( s ) . CASTRYCK, CLUCKERS, DITTMANN, AND NGUYEN
Corollary (Main Lemma 2.5 of [25]) . Let
X →
Spec Z be the hypersurface in P n Z cut out by the homogeneous polynomial f as above, so X is the generic fibre of X and X p is the special fibre of X over p .Let P be an F p -point of multiplicity µ on X p and let ξ , . . . , ξ s be Z -points on X ,given by some primitive integer tuples, with reduction P . Let F , . . . , F s be homoge-neous polynomials in x , . . . , x n with integer coefficients.Then det (cid:0) F j ( ξ i ) (cid:1) is divisible by p e where e ≥ ( n ! /µ ) /n nn + 1 s /n − O n ( s ) . Proof.
Let P ′ be the image of P under the closed embedding X p ֒ → X , and R thestalk of X at P ′ . Then R is a noetherian local ring containing Z ( p ) , and R/pR isthe stalk of X p at P . Since P ′ is a specialization of all the ξ i (this is precisely whatit means that the ξ i have reduction P ), it makes sense to evaluate an element of R at each ξ i , giving s ring homomorphisms R → Z ( p ) .The F i induce Z ( p ) -valued polynomial functions on an affine neighbourhood of P ′ ,and hence give elements of R . The statement now follows from the preceding twolemmas. (cid:3) Proposition.
Let X be as above. Let ξ , . . . , ξ s be Z -points on X , and F , . . . , F s be homogeneous polynomials in n + 1 variables with integer coefficients. Then thedeterminant ∆ of the s × s -matrix ( F i ( ξ j )) is divisible by p e , where e ≥ ( n !) /n nn + 1 s /n n /np − O n ( s ) , and where n p is the number of F p -points on X p , counted with multiplicity.Proof. This is identical to the proof of [26, Lemma 1.4], see also the appendix of [30]– but we have eliminated the dependence on the constant on d . (cid:3) Lemma.
In the situation above, if p > d and X p is geometrically integral,i.e. the defining polynomial f has absolutely irreducible reduction modulo p , then n p ≤ p n + O n ( d p n − / ) .Proof. By [9, Corollary 5.6] the number of F p -points of X p counted without multi-plicity is bounded by p n +1 + ( d − d − p n +1 / + (5 d + d + 1) p n − p − ≤ p n + O n ( d p n − / ) . (This uses the lower bound on p and the condition on X p .)The singular points of X p all lie in the algebraic set cut out by f and ∂f∂x , which canbe assumed non-zero without loss of generality. This is an algebraic set all of whosecomponents have codimension and the sum of the degrees of these components isbounded by d . The standard Lang–Weil estimate yields that there are O n ( d p n − ) ≤ O n ( dp n − / ) points on this algebraic set and hence at most that many singular points,each of which has multiplicity at most d . Adding this term to the number of pointscounted without multiplicity yields the claim. (cid:3) HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 9
Lemma.
In the situation above, with p > d and X p geometrically integral,we have n /np /p − ≤ O n ( d p − / ) .Proof. Apply the general inequality x /n − ≤ x − for x ≥ . (cid:3) We immediately obtain the following from Proposition 2.6.2.9.
Corollary.
The determinant ∆ from Proposition 2.6 is divisible by p e , where e ≥ ( n !) /n nn + 1 s /n p + O n ( d p / ) − O n ( s ) . This is stated as Theorem 2.2 in [30], but our statement is more precise in termsof the implied constants.3.
Points on projective hypersurfaces à la Walsh
Formulation of main result.
The following result is the goal of this sectionand an improvement to Theorem 1.3 of [30]. Call a polynomial f over Z primitiveif the greatest common divisor of its coefficients equals . For any f , we write k f k for the maximum of the absolute values of the coefficients of f .3.1.1. Theorem.
Let n > be an integer. Then there exists c such that the followingholds for all choices of f, d, B . Let f be a primitive irreducible homogeneous poly-nomial in Z [ x , . . . , x n +1 ] of degree d ≥ , and write X for the hypersurface in P n +1 Q cut out by f . Choose B ≥ . Then there exists a homogeneous g in Z [ x , . . . , x n +1 ] of degree at most cB n +1 nd /n d − /n b ( f ) k f k n d /n + cd − /n log B + cd − /n , not divisible by f , and vanishing at all points on X of height at most B . Here the quantity b ( f ) will be defined in Definition 3.2.1; it always satisfies b ( f ) ≤ O (max( d − log k f k , . The main improvement over [30] lies in the poly-nomial dependence on the degree d .We also immediately obtain the following, which is the essential tool for provingTheorem 2.3.1.2. Corollary.
For any primitive irreducible polynomial f ∈ Z [ x , x , x ] homo-geneous of degree d and any B ≥ we have N ( f, B ) ≤ cB d d b ( f ) k f k /d + cd log B + cd ≤ c ′ d B d , where c , c ′ are absolute constants.Proof. Apply Theorem 3.1.1 to obtain a polynomial g , and then apply Bézout’stheorem to the curves defined by f and g . This yields the first inequality. Forthe second inequality we can use that b ( f ) / k f k /d is bounded because b ( f ) ≤ O (max( d − log k f k , . (cid:3) A determinant estimate.
In this section we want to use the results of Section2 for a number of primes simultaneously. It is useful to introduce the followingmeasure of the set of primes modulo which an absolutely irreducible polynomialover the integers ceases to be absolutely irreducible.3.2.1.
Definition.
For an integer polynomial f in an arbitrary number of variableswe set b ( f ) = 0 if f is not absolutely irreducible, and b ( f ) = Y p exp( log pp ) otherwise, where the product is over those primes p > d such that the reductionof f modulo p is not absolutely irreducible.For now we work with a degree d hypersurface in P n +1 defined by a primitivepolynomial f ∈ Z [ x , . . . , x n +1 ] which is absolutely irreducible. We first establish abasic estimate on b ( f ) , showing in particular that it is finite.3.2.2. Theorem (Explicit Noether polynomials, [24, Satz 4]) . Let d ≥ , n ≥ .There is a collection of homogeneous polynomials Φ in (cid:0) n + dn (cid:1) variables over Z ofdegree d − , such that k Φ k ≤ d d − (cid:2)(cid:18) n + dn (cid:19) d (cid:3) d − (where k·k denotes the sum of the absolute values of the coefficients), and such thatthe following holds for any polynomial F in n + 1 variables homogeneous of degree d over any field: • if F is not absolutely irreducible, then all Φ ’s vanish when applied to thecoefficients of F , reducing modulo the characteristic of the ground field ifnecessary; • if F is absolutely irreducible over a field of characteristic , then one of the Φ ’s does not vanish when applied to the coefficients of F . Corollary. b ( f ) ≤ O (max( d − log k f k , .Proof. Write P for the set of prime numbers p > d modulo which f is notabsolutely irreducible. There exists a Noether form Φ with coefficients in Z suchthat Φ applied to the coefficients of f is non-zero, but is divisible by any prime in P . In particular, the product of such p is bounded by c := k Φ k k f k degΦ . Now log b ( f ) = X p ∈P log pp ≤ X d
0) + O (1) + log c log c ≤ max(log log c − d,
0) + O (1) ≤ max(log(degΦ log k f k ) − d, log log k Φ k − d,
0) + O (1) , HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 11 where we have used that the function log x − P p ≤ x log pp is bounded (Mertens’ firsttheorem). Since log log k Φ k − d is bounded above, the claim follows. (cid:3) We now adapt [30, Theorem 2.3], keeping track of the dependency on the degreeand on b ( f ) .3.2.4. Lemma.
For any x > , P p ≤ x log p ≤ x , where the sum extends over primenumbers not exceeding x .Proof. This is a classical estimate on the first Chebyshev function. (cid:3)
Lemma. As x varies over positive real numbers we have P p>x log pp / = O ( x − / ) ,where the sum extends over prime numbers greater than x .Proof. Estimate the density of prime numbers using the prime number theorem andcompare the sum with an integral. (cid:3)
Proposition.
Let ( ξ , . . . , ξ s ) be a tuple of rational points in X , let F li ∈ Z [ x , . . . , x n +1 ] , ≤ l ≤ L , ≤ i ≤ s , be homogeneous polynomials with integercoefficients, and write ∆ l for the determinant of ( F li ( ξ j )) ij . Let ∆ be the greatestcommon divisor of the ∆ l , and assume that ∆ = 0 . Then we have the bound log | ∆ | ≥ n ! /n n + 1 s /n (log s − O n (1) − n (4 log d + log b ( f ))) . This is a more explicit variant of [30, Theorem 2.3].
Proof.
Let P be the collection of prime numbers p such that either p ≤ d or X p is not geometrically integral.We now apply Corollary 2.9 to all prime numbers p ≤ s /n not in P , yielding log | ∆ | ≥ n ! /n nn + 1 s /n X P6∋ p ≤ s /n log pp + O n ( d p / ) − O n ( s ) X p ≤ s /n log p. The last term is bounded by O n (1) s /n .In estimating the main term, we may use that p + O n ( d p / ) ≥ p − O n ( d ) p / . Wecan then bound X P6∋ p ≤ s /n log pp + O n ( d p / ) ≥ X p ≤ s /n log pp − X p ∈P log pp − O n ( d ) X P6∋ p ≤ s /n log pp / ≥ log sn − X p ≤ d log pp − log b ( f ) − O (1) − O n ( d X p> d log pp / ) ≥ log sn − log(27 d ) − log b ( f ) − O (1) − O n ( d (27 d ) − / ) ≥ log sn − d − log b ( f ) − O n (1) . (cid:3) The main estimates.
We first establish that we can reduce to the case ofabsolutely irreducible f in the proof of Theorem 3.1.1.3.3.1. Lemma. If f ∈ Z [ x , . . . , x n +1 ] is homogeneous of degree d ≥ and ir-reducible but not absolutely irreducible, then there exists another polynomial g ∈ Z [ x , . . . , x n +1 ] of degree d , not divisible by f , which vanishes on all rational zeroesof f .Proof. This is established in the first paragraph of Section 4 of [30]. (cid:3)
Let us now work with a restricted class of homogeneous polynomials f , namelythose which are absolutely irreducible and for which the leading coefficient c f , i.e. thecoefficient of the monomial x dn +1 , satisfies c f ≥ k f k C − nd /n for some positive constant C which is allowed to depend on n (for this reason thefactor n in the exponent is in fact superfluous, but it simplifies the proof write-upbelow).The two main results are the following.3.3.2. Lemma.
For f as above, and B satisfying k f k ≤ B d ( n +1) , there exists ahomogeneous polynomial g not divisible by f , vanishing at all zeroes of f of heightat most B , and of degree M = O n (1) B n +1 nd /n d − /n b ( f ) k f k n − d − − /n + d − /n log B + O n ( d ) . Lemma.
For f as above, and B satisfying k f k ≥ B d ( n +1) , there exists ahomogeneous polynomial g not divisible by f , vanishing at all zeroes of f of heightat most B , and of degree M = O n ( d − /n ) . These two lemmas together clearly imply the statement of Theorem 3.1.1, at leastfor polynomials f satisfying the condition on leading coefficients.We follow the exposition in [30, Section 4], and prove the two lemmas together.We shall need the following.3.3.4. Theorem ([4, Theorem 1]) . Let P rk =1 a mk x k = 0 ( m = 1 , . . . , s ) be a systemof s linearly independent equations in r > s variables x , . . . , x r , with coefficients a mk ∈ Z . Then there exists a non-trivial integer solution ( x , . . . , x r ) satisfying max ≤ i ≤ r | x i | ≤ ( D − p | det( AA ⊤ ) | ) r − s . Here A = ( a mk ) is the matrix of coefficients and D is the greatest common divisorof the determinants of the s × s minors of A .Proof of Lemmas 3.3.2 and 3.3.3. Fix B ≥ , and let S be the set of rational pointson the hypersurface described by f of height at most B . Let M > be suchthat there is no homogeneous polynomial g of degree M , not divisible by f , which HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 13 vanishes on all points in S ; we shall show that M is bounded in terms of n, B, d, k f k as stated. Let us assume in the following that M is bigger than some constant (tobe specified later) times d .Given an integer D , write B [ D ] for the set of monomials of degree D in n +2 variables, so |B [ D ] | = (cid:0) D + n +1 n +1 (cid:1) . Write Ξ ⊆ S for a maximal subset which isalgebraically independent over monomials of degree M , in the sense that applyingall monomials in B [ M ] to Ξ yields s = | Ξ | linearly independent vectors. Let A bethe s × r matrix whose rows are these vectors, where r = |B [ M ] | = (cid:0) M + n +1 n +1 (cid:1) ; eachentry of A is bounded in absolute value by B M . Since all polynomials in f · B [ M − d ] vanish on Ξ , and no polynomials of degree M not divisible by f do by assumptionon M , we have s = |B [ M ] | − |B [ M − d ] | .Now A describes a system of linear equations whose solutions correspond to (thecoefficients of) homogeneous polynomials of degree M vanishing on all points in Ξ and therefore all points in S ; by assumption, these polynomials are multiplesof f and therefore have one coefficient of size at least c f ≥ k f k C − nd /n by theassumption on f . Hence Theorem 3.3.4 yields ∆ ≤ p | det( AA ⊤ ) | ( k f k C − nd /n ) s − r , where we write ∆ for the greatest common divisor of the determinants of the s × s minors of A . Taking logarithms, using the estimate | det( AA ⊤ ) | ≤ s !( rB M ) s obtainedby estimating the size of the coefficients of AA ⊤ , and using the estimate for ∆ obtained from Proposition 3.2.6, this expands as follows: n ! /n n + 1 s /n (log s − O n (1) − n (4 log d + log b ( f ))) ≤ log s !2 + s r + sM log B − ( r − s ) (cid:0) log k f k − nd /n O n (1) (cid:1) We can use the estimates log s ! ≤ s log s and log r ≤ log( M +1) n +1 ≤ O n (log M ) ≤ O n (log s ) to see that the first two terms on the right-hand side are both in O n ( s /n ) and can hence be neglected by adjusting the constant O n (1) on the left-hand side.Diving by M s now yields:(3.3.1) n ! /n n + 1 s /n M (log s − O n (1) − n (4 log d + log b ( f ))) ≤ log B − r − sM s (cid:0) log k f k − nd /n O n (1) (cid:1) The term s = (cid:0) M + n +1 n +1 (cid:1) − (cid:0) M − d + n +1 n +1 (cid:1) is a polynomial in M and d . We can write s = dM n n ! + O n ( d M n − ) , in particular log s = log d + n log M − O n (1) . By rearranging and applying thebinomial series, which is legal since d /M is bounded above by an adjustable absolute constant, we also obtain s /n M = d /n n ! /n + O n ( d M ) . Thus the left-hand side of the inequality above can be replaced by d /n nn + 1 (log M − O n (1) − ((4 − /n ) log d + (1 + O n ( d − /n M )) log b ( f ))) , where we have dropped terms O n ( d − /n log M/M ) and O n ( d − /n log d/M ) by ad-justing the constant in O n (1) .Let us now estimate r − sMs . We have r − s = M n +1 ( n +1)! + O n ( dM n ) , so r − sM s = 1 d ( n + 1) 1 + O n ( d/M )1 + O n ( d/M ) = 1 d ( n + 1) + O n ( 1 M ) . Therefore inequality (3.3.1) becomes(3.3.2) d /n nn + 1 (log M − O n (1) − ((4 − /n ) log d + (1 + O n ( d − /n M )) log b ( f ))) ≤ log B − log k f k d ( n + 1) − O n ( log k f k M ) . Let us now assume that k f k ≤ B d ( n +1) and M ≥ d − /n log B . Then log k f k ≤ d ( n + 1) log B ≤ O n ( d /n M ) , so we can drop the last term on the right-hand side,as well as the O n ( log b ( f ) M ) on the left-hand side. Rearranging yields that log M ≤ O n (1) + n + 1 d /n n log B − log k f k nd /n + (4 − /n ) log d + log b ( f ) , so we obtain Lemma 3.3.2.Now, on the other hand, assume that k f k ≥ B d ( n +1) and M ≥ d ( n + 1) .Rearranging inequality (3.3.2) yields log M ≤ O n (1) + (4 − /n ) log d + (1 + O n ( d − /n )) log b ( f ) − log k f k nd /n ≤ O n (1) + max { d, (4 − /n ) log d } , where we have used O n (1) log b ( f ) − log k f k nd /n ≤ O n (1) max(log log k f k − d, − log k f k nd /n ≤ max(0 , − d + log O n (1)4 nd /n ) ≤ O n (1) by Corollary 3.2.3 and the lemma below. This establishes Lemma 3.3.3. (cid:3) Lemma.
Let c > . For any x > we have log log x − log x/c ≤ log c + O (1) . HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 15
Proof.
Let C = sup x> (log log x − log x ) ; note that the supremum exists, since it istaken over a continuous function on ]1 , ∞ [ which tends to −∞ at both ends of theinterval. Now log log x − log x/c = log c + log log x /c − log x /c ≤ log c + C . (cid:3) Finishing the proof.
We use the ideas from [30, Section 3] to finish the proofof Theorem 3.1.1.3.4.1.
Lemma.
Let f ∈ C [ x ] be a polynomial of degree ≤ d , and write k f k for themaximal absolute value among the coefficients. There exists an integer a , ≤ a ≤ d ,such that | f ( a ) | ≥ − d k f k .Proof. This is a statement about the k·k ∞ -operator norm of the inverse of the Van-dermonde matrix with nodes , . . . , d , which can be deduced from [13, Theorem1]. (cid:3) Lemma.
Let f ∈ C [ x , . . . , x n +1 ] be homogeneous of degree d . There existintegers a , . . . , a n with ≤ a i ≤ d such that | f ( a , . . . , a n , |≥ − ( n +1) d k f k .Proof. Dehomogenize by setting x n +1 = 1 , and then use induction with the precedinglemma. (cid:3) Proof of Theorem 3.1.1.
Take a non-zero f ∈ Z [ x , . . . , x n +1 ] homogeneous of degree d . Consider a , . . . , a n as in the last lemma and let A = I + A ∈ SL n +2 ( Z ) , where I is the ( n + 2) × ( n + 2) identity matrix and A has its last column equal to ( a , . . . , a n , and zero everywhere else. Note that A − = I − A .Let f ′ = f ◦ A . By construction, the x dn +1 -coefficient of f ′ is ≥ − ( n +1) d k f k .Because of the boundedness of the entries of A , we furthermore see that k f ′ k ≤ (cid:18) n + d + 1 n + 1 (cid:19) d n +1 k f k . In particular, the x dn +1 -coefficient of f ′ is greater than C − nd /n k f ′ k for some con-stant C depending only on n . The polynomial f ′ is primitive if and only if f is, sincethey are related by the matrices A , A − with integer coefficients, and b ( f ) = b ( f ′ ) .Furthermore, if g ′ is a homogeneous polynomial in Z [ x , . . . , x n +1 ] vanishing on allzeroes of f ′ up to a certain height B ′ , then g = g ′ ◦ A − is a polynomial of the samedegree vanishing on all zeroes of f up to height B = B ′ / ( d + 1) .Since either Lemma 3.3.2 or Lemma 3.3.3 applies to f ′ and B ′ , we obtain thedesired statement for f . (cid:3) Proofs of Theorems 1, 2, 3, 4, 6
On trivial bounds.
In this subsection, we extend our notation to varietiesdefined over any field K containing Q , and we write N ( X, B ) for the number ofpoints in P n ( Q ) ∩ X ( K ) of height at most B , when X is a subvariety of P nK , andsimilarly we write N aff ( Y, B ) for the number of points in Z n ∩ Y ( K ) ∩ [ − B, B ] n ,when Y ⊆ A nK . Lemma.
Let X ⊆ A n Q be a (possibly reducible) variety of pure dimension m and degree d defined over Q . Then the number N aff ( X, B ) of integral points on X of height at most B is bounded by d (2 B + 1) m . When X is a hypersurface, this is the well-known Schwarz-Zippel bound, andeven the general case appears in many places in the literature, albeit often withoutmaking the bound completely explicit. Proof.
This is an easy inductive argument using intersections with shifts of coor-dinate hyperplanes. In fact, the proof of [6, Theorem 1] automatically gives thisstronger statement. (cid:3)
Corollary.
For an irreducible affine variety X in A n of degree d and dimen-sion < n there exists a tuple ( a , . . . , a n ) of integers not on X , with | a i | ≤ d forevery i . For every irreducible projective variety X in P n of degree d and dimension < n there exists a point in P n ( Q ) of height at most d not on X .Proof. The affine version is implied by the preceding lemma, and the projectiveversion follows by considering the affine cone. (cid:3)
Lemma.
Let X ⊆ A n Q be an absolutely irreducible variety of dimension m anddegree d not defined over Q . Then the number N aff ( X, B ) of integral points on X ofheight at most B is bounded by d (2 B + 1) m − . By considering the affine cone over a projective variety, this result also applies toprojective varieties of dimension m , with bound d (2 B + 1) m . Proof.
For every field automorphism σ of Q , there is a conjugate variety X σ . Since X is not defined over Q , there exists a σ with X σ = X . All Q -points of X necessarilyalso lie on X σ . Since X σ has degree d , it is the intersection of hypersurfaces of degree ≤ d , see for instance [18, Proposition 3]. Let Y be a hypersurface of degree ≤ d containing X σ and not containing X . Then X ∩ Y is a variety of pure dimension m − and degree at most d . Now Lemma 4.1.1 gives the result. (cid:3) The following allows us to reduce to the geometrically irreducible situation whencounting points on varieties.4.1.4.
Corollary.
Let X ⊆ A n be an irreducible variety over Q of dimension m and degree d which is not geometrically irreducible. Then for any B ≥ we have N aff ( X, B ) ≤ d (2 B + 1) m − . As above, this also applies to projective varieties.
Proof.
Let K/ Q be a finite Galois extension over which X splits into absolutelyirreducible components, and let Y be one of the components. Since all componentsare Galois-conjugate, the Q -points on X in fact also lie on Y . Now the precedinglemma applied to Y gives the result. (cid:3) HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 17
Remark.
Note that this trivially proves Theorems 1 and 3 for irreducible, butnot geometrically irreducible varieties, and similarly for absolutely irreducible vari-eties defined over Q but not over Q . The same applies for Theorem 2 by consideringa projective curve as the union of an affine curve with a finite number of points.Thus we henceforth only need to concern ourselves with absolutely irreduciblevarieties defined over Q .4.2. Affine counting.
Our results for projective hypersurfaces from the last sectionyield the following result for affine hypersurfaces, by refining the technique given in[11, Remark 2.3].4.2.1.
Proposition.
Fix an integer n > . Then there exist c and e such that thefollowing holds for all f, B, d . Let f be in Z [ x , . . . , x n +1 ] be irreducible, primitiveand of degree d . For each i write f i for the degree i homogeneous part of f . Fix B ≥ . Then there is a polynomial g in Z [ x , . . . , x n +1 ] of degree at most cB d /n d − /n min(log k f d k + d log B + d , d b ( f )) k f d k n d /n + cd − /n log B + cd − /n , not divisible by f , and vanishing on all points x in Z n +1 satisfying f ( x ) = 0 and | x i | ≤ B . To prove Proposition 4.2.1 we need the following lemmas.4.2.2.
Lemma ([7, Lemma 5]) . Let f ∈ Z [ x , . . . , x n +2 ] be a primitive absolutely irre-ducible polynomial, homogeneous of degree d , defining a hypersurface Z in P n +1 . Let B ≥ . Then either the height of the coefficients of f is bounded by O n ( B d ( d + n +1 n +1 )) ,or there exists a homogeneous polynomial g of degree d vanishing on all points of Z of height at most B . Lemma.
For F ∈ Z [ x , . . . , x n +2 ] an irreducible primitive homogeneous poly-nomial and ≤ y ≤ k F k we have d − /n b ( F ) k F k n d /n ≤ O n (1) d − /n log y + d y n d /n . Proof.
The function x log xx n d /n on (1 , ∞ ) is monotonically increasing up to its maximum when x n d /n = e , andmonotonically decreasing thereafter.Let us write x = k F k and use d b ( F ) ≤ O n (1)(log x + d ) by Corollary 3.2.3.By the monotonicity considered above, there is nothing to show when y n d /n ≥ e .Otherwise, d − /n log y + d y n d /n ≥ d − /n d y n d /n ≥ d − /n ( 1 e + 1 y n d /n ) , and the left-hand side of the inequality in the statement is always bounded by O n (1) d − /n log x + d x n d /n ≤ O n (1) d − /n ( nd /n e + d x n d /n ) , yielding the claim. (cid:3) Proof of Proposition 4.2.1.
By applying Lemma 3.3.1 to the homogenization of f , wemay assume that f is absolutely irreducible. For each natural number H , considerthe polynomial F H ∈ Z [ x , ..., x n +2 ] given by F H ( x , ..., x n +2 ) = P di =0 H i f i x d − in +2 .Then F H is an irreducible homogeneous polynomial of degree d . On the other hand,each integral point ( x , ..., x n +1 ) ∈ Z ( f )( Z ) gives us a rational point ( x , ..., x n +1 , H ) in Z ( F H )( Q ) , where Z ( f ) stands for the hypersurface in A n +1 given by f and Z ( F H ) stands for the hypersurface in P n +1 given by F H .If B is bounded by some polynomial expression in d (to be determined later),then B nd /n is bounded by a constant depending only on n ; hence we use Theorem3.1.1 for F , by which there exists a number c depending only on n along with ahomogeneous polynomial G in Z [ x , . . . , x n +2 ] of degree at most cB d /n d − /n b ( F ) k F k n d /n + cd − /n log B + cd − /n , not divisible by F , and vanishing at all points on Z ( F )( Q ) of height at most B .Since b ( F ) = b ( f ) and k F k ≥ k f d k , by Lemma 4.2.3 we obtain d − /n b ( F ) k F k n d /n ≤ O n ( d − /n ) min( d b ( f ) , log k f k + d ) k f d k n d /n . Hence the polynomial g ( x , ..., x n +1 ) = G ( x , ..., x n +1 , satisfies our proposition.For any B ≥ Bertrand’s postulate guarantees the existence of a prime B ′ inthe interval ( B , B ] . Moreover, if B ′ ∤ f , then F B ′ is primitive. By Theorem 3.1.1for F B ′ , there exists a number c depending only on n along with a homogeneouspolynomial G B ′ in Z [ x , . . . , x n +2 ] of degree at most cB n +1 nd /n d − /n b ( F B ′ ) k F B ′ k n d /n + cd − /n log B + cd − /n , not divisible by F B ′ , and vanishing at all points on Z ( F B ′ )( Q ) of height at most B .It is clear that k F B ′ k ≥ B ′ d k f d k ≥ − d B d k f d k , so by Lemma 4.2.3 we have d − /n b ( F B ′ ) k F B ′ k n d /n ≤ O n (1)( B − nd /n d − /n log k f d k + d log B + d k f d k n d /n . Furthermore b ( F B ′ ) agrees with b ( F ) up to a factor of exp(log B ′ /B ′ ) ≤ O (1) . Hencewe in fact have d − /n b ( F B ′ ) k F B ′ k n d /n ≤ O n (1) B − nd /n d − /n min(log k f d k + d log B + d , b ( f )) k f d k n d /n . Thus the polynomial g ( x , ..., x n +1 ) = G B ′ ( x , ..., x n +1 , B ′ ) is as desired. HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 19
From now on, we suppose that
B > and B ′ | f for all primes B ′ in the interval ( B , B ] . Then we have ( Y B ′ prime B/
There exists a constant c such that for all d > and all irreducibleaffine curves X ⊆ A Q of degree d , cut out by an irreducible primitive polynomial f ∈ Z [ x , x ] , and all B ≥ one has N aff ( X, B ) ≤ cB /d min( d log k f d k + d log B + d , d b ( f )) k f d k /d + cd log B + cd . Proof.
Take n = 1 in Proposition 4.2.1 and apply Bézout’s theorem. (cid:3) If the absolute irreducibility of f can be explained by the indecomposability ofits Newton polytope, e.g., in the sense of [12], then this allows for good bounds on b ( f ) which get rid of the factor log B . The following instance will be used to proveTheorem 6:4.2.5. Corollary.
There exists a constant c such that for all affine curves X ⊆ A Q cut out by a polynomial f ∈ Z [ x , x ] of the form c d x d + c d ′ x d ′ + X i,i ′ id ′ + i ′ d
By dividing out by the greatest common divisor of the coefficients, we maysuppose that f is primitive. The presence of the edge ( d, – (0 , d ′ ) in the Newtonpolytope of f is enough to guarantee absolute irreducibility in any characteristic [12,Theorem 4.11]. Therefore we can bound b ( f ) ≤ Y p | c d c d ′ exp( log pp ) ≤ log | c d c d ′ | + 1 through Mertens’ first theorem as in Corollary 3.2.3. (cid:3) Proofs of our main results.
We can now prove our main theorems, subjectto the following propositions; they allow us to reduce to the case of hypersurfacesthroughout, and will be established in Section 5 by projection arguments.4.3.1.
Proposition.
Given a geometrically integral affine variety X in A n of dimen-sion m and degree d , there exists a geometrically integral affine variety X ′ in A m +1 birational to X , also of degree d , such that for any B ≥ we have N aff ( X, B ) ≤ dN aff ( X ′ , c n d e n B ) , where c n , e n are constants depending only on n .For m = 1 , we can even achieve N aff ( X, B ) ≤ N aff ( X ′ , c n d e n B ) + d . Proposition.
Given a geometrically integral projective variety X in P n ofdimension m and degree d , there exists a geometrically integral projective variety X ′ in P m +1 birational to X , also of degree d , such that for any B ≥ we have N ( X, B ) ≤ dN ( X ′ , c n d e n B ) , where c n , e n are constants depending only on n .For m = 1 , we can even achieve N ( X, B ) ≤ N ( X ′ , c n d e n B ) + d . Proof of Theorem 2.
In the case of a planar curve, i.e. for n = 2 , Corollary 3.1.2 givesthe claim. For the general case, we may assume that the given curve is geometricallyintegral by Remark 4.1.5, and then reduce to n = 2 by applying Proposition 4.3.2(where m = 1 ). (cid:3) Proof of Theorem 3.
We may assume that the curve X is geometrically integral byRemark 4.1.5. In the case of a planar curve, i.e. for n = 2 , Corollary 4.2.4 yieldsthat N ( X, B ) ≤ O n (( d log B + d ) B /d ) , by observing that d log k f d k + d log B + d k f d k /d ≤ d log B + 2 d . HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 21
We can reduce the general case to n = 2 by applying Proposition 4.3.1 (where m = 1 ), yielding the same estimate. (cid:3) Proof of Theorem 6.
In the penultimate step of their proof of [1, Theorem 1.1], Bhar-gava et al. establish the bound h ( K ) ≤ O d,ε ( | ∆ K | + ε ) + X β ∈B N aff ( f β , | ∆ K | ) where B ⊆ O K is a set of size O d ( | ∆ K | − d ) and f β = y − N K/ Q ( x − β ) = y − x d − lower order terms in x. Theorem 3 implies that N aff ( f β , | ∆ K | ) ≤ O d ( | ∆ K | d log | ∆ K | ) , yielding the desired result when d is even. If d is odd then instead of Theorem 3we apply Corollary 4.2.5 with d ′ = 2 , c d = − , c d ′ = 1 to get rid of the factor log | ∆ K | . (cid:3) For the proof of Theorem 4, we need the following explicit form of Proposition 1of [7] with D = 1 .4.3.3. Proposition.
There exists a constant c such that for all d ≥ and all poly-nomials f ∈ Z [ x , x , x ] of degree d such that the highest degree part h ( f ) = f d of f is irreducible, all finite sets I of curves C of A Q of degree and lying on thehypersurface defined by f , and all B ≥ one has N aff ( X ∩ ( ∪ C ∈ I C ) , B ) ≤ cd B + I. Proof.
We write I = I ∪ I where I = { L ∈ I | N aff ( L, B ) ≤ } and I = { L ∈ I | N aff ( L, B ) > } . It is clear that N aff ( X ∩ ∪ L ∈ I L ) ≤ I . If L ∈ I , then thereexist a = ( a , a , a ) , v = ( v , v , v ) ∈ Z such that H ( a ) ≤ B , v is primitive and L ( Q ) = { a + λv | λ ∈ Q } . Since v is primitive we deduce that L ( Z ) ∩ [ − B, B ] = { a + λv | λ ∈ Z , H ( a + λv ) ≤ B } . So L ( Z ) ∩ [ − B, B ] ) ≤ BH ( v ) . Since L ∈ I we have H ( v ) ≤ B and f d ( v ) = 0 . On the other hand, for eachpoint v with f d ( v ) = 0 , there are at most d ( d − lines L ∈ I in the directionof v , since each such line intersects a generic hyperplane in A in a point which issimultaneously a zero of f and of the directional derivative of f in the direction of v . Put A i = { v ∈ P ( Q ) | f d ( v ) = 0 , H ( v ) = i } and n i = A i . Then, by Corollary3.1.2, there exists a constant c independent of f such that P ≤ i ≤ k n i ≤ cd k d . Byour discussion, N aff ( X ∩ ( ∪ C ∈ I C ) , B ) ≤ I + ( d − d B X i =1 n i (1 + 2 Bi ) . On the other hand, summation by parts gives the following: B X i =1 n i (1 + 2 Bi ) = B − X k =1 ( k X i =1 n i )( 2 Bk − Bk + 1 ) + ( B X i =1 n i )(1 + 2 B B ) ≤ cd ( B − X k =1 k d Bk ( k + 1) + 2(2 B ) d ) . Since d ≥ , one has P k ≥ k d k ( k +1) < + ∞ and B d ≤ B . Thus, by enlarging c , wehave N aff ( X ∩ ( ∪ C ∈ I C ) , B ) ≤ cd B + I as desired. (cid:3) In order to prove Theorem 4, we now first consider the case of a surface in P ,with proof inspired by the proof of Corollary 7.3 of [26] in combination with theimprovements developed above.4.3.4. Proposition.
There exists a constant c such that for all polynomials f in Z [ y , y , y ] whose homogeneous part of highest degree f d is irreducible over Q andwhose degree d is least , one has N aff ( f, B ) ≤ cd B .Proof of Proposition 4.3.4 for d ≥ . For any prime modulo which f d is absolutelyirreducible, the reduction of f is likewise absolutely irreducible, so b ( f ) ≤ b ( f d ) .Applying the usual estimate from Corollary 3.2.3, Proposition 4.2.1 yields for each B ≥ a polynomial g of degree at most(4.3.1) cd / B / √ d , not divisible by f and vanishing on all points x in Z n satisfying f ( x ) = 0 and | x i | ≤ B , with c an absolute constant. Let C be an irreducible component of the(reduced) intersection of f = 0 with g = 0 . Call this intersection C . If C is of degree δ > , then(4.3.2) N aff ( C, B ) ≤ c ′ δ B /δ (log B + δ ) by Theorem 3, for some absolute constant c ′ .By Proposition 4.3.3, the total contribution of integral curves D of C of degree is at most(4.3.3) c ′′ d B for some absolute constant c ′′ .Suppose that C , ..., C k are irreducible components of the intersection of f = 0 and g = 0 and deg( C i ) > for all i . Furthermore, we assume that deg( C i ) ≤ log B for all ≤ i ≤ m and deg( C i ) > log B for all i > m . Since the function δ B ( δ ) + δ isdecreasing in (0 , log B ) and increasing in ( log B , + ∞ ) , by enlarging c ′ , for all ≤ i ≤ m we have(4.3.4) N aff ( C i , B ) ≤ c ′ B / (log B + 1) . HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 23
On the other hand, if δ > log B then B δ is bounded, so (4.3.1) and (4.3.2) imply(4.3.5) X m +1 ≤ i ≤ k N aff ( C i , B ) ≤ c ′′′ d B / √ d for some c ′′′ independent of d and B .Putting the estimates (4.3.1), (4.3.3), (4.3.4), (4.3.5) together proves the propo-sition when d is at least . (cid:3) To give a proof of Proposition 4.3.4 for lower values of d than , one could tryto get a form of Theorem 3 with a lower exponent of the degree and repeat theabove proof. We proceed differently: we treat the values for d going from up to by inspecting the proof of [7, Theorem 2] in combination with some of the aboverefinements, and the case of d = 5 by using [26, Theorem 7.2] (at the cost of beingless self-contained). Proof of Proposition 4.3.4 with ≤ d ≤ . Fix ≤ d ≤ , let f ∈ Z [ y , y , y ] beof degree d with absolutely irreducible homogeneous part of highest degree, and let X be the surface described by f .In [7, Theorem 2], the estimate N aff ( f, B ) ≤ O d,ε ( B ε ) is established for every ε > . However, using our Theorem 2 and Proposition 4.3.3, we shall show thattheir proof [7, pp. 568–570] in fact gives the bound N aff ( f, B ) ≤ O d ( B ) , withoutany ε , which is sufficient for our purposes.Specifically, they first consider the case in which Lemma 4.2.2 applies, so all therational points on X of height up to B lie on a union of irreducible curves withsum of degrees at most d . Applying Theorem 2 to those curves of degree ≥ andProposition 4.3.3 for the contribution of curves of degree yields the claim in thiscase.In the remaining case, it is argued that there is an open subset U ⊆ X (specificallyconsisting of those nonsingular points on X which have multiplicity at most onthe tangent plane section at the point) whose complement consists of O d (1) integralcomponents of degree O d (1) ; by the same argument as in the preceding paragraph,the contribution of this complement is O d ( B ) , so it suffices to estimate N aff ( U, B ) .Further, it is argued that the points on U of height at most B are covered bya certain collection of irreducible curves. The subcollection I consisting of thosecurves of degree at most satisfies | I | ≤ O d,ε ( B / √ d +2 ε ) , so our Proposition 4.3.3and [7, Proposition 1] gives a contribution O d,ε ( B + B / √ d +3 ε ) ≤ O d ( B ) .The remaining curves, of which there are no more than O d,ε ( B / √ d ) , all contributeat most B / − / (2 √ d ) ([7, Proposition 2]), so their total contribution is O d,ε ( B / (2 √ d )+1 / ε ) ≤ O d ( B ) . (cid:3) Theorem ([26, Theorem 7.2]) . Let X be a geometrically integral surface in P Q of degree d and X ns its non-singular locus. Suppose that the hyperplane definedby x = 0 intersects X properly, and identify A with the open subset of P givenby x = 0 . There exists a positive constant c bounded solely in terms of d such that the following holds: for every B ≥ there exists a set of O d ( B / √ d log B + 1) geometrically integral curves D λ on X of degree O d (1) such that N aff ( X ns \ [ λ D λ , B ) ≤ O d ( B / √ d + c/ log(1+log B ) ) . Proof of Proposition 4.3.4 for d = 5 . Suppose that the degree d of f is exactly ,and let X be the surface in A Q given by f . We may assume that B ≥ . ByTheorem 4.3.5, there is c > such that for each B ≥ there is a set C of at most cB / √ d log B geometrically integral curves C ⊆ A Q of degree at most c and lying on X such that N aff ( X ns \ [ C ∈C C, B ) ≤ O ( B / √ d + c/ log(log B ) ) ≤ O ( B / ) , where X ns is the open subvariety of nonsingular points.The complement of X ns in X is a union of irreducible curves the sum of whosedegrees is bounded by a constant. Applying Theorem 2 to those curves of degree ≥ and Proposition 4.3.3 for the contribution of curves of degree yields that thecomplement of X ns contributes at most O ( B ) points, which is satisfactory for ourpurposes.Similarly, the curves in C of degree contribute at most O ( cB / √ d log B + B ) ≤ O ( B ) points by Proposition 4.3.3, and the curves in C of degree ≥ each contributeat most O ( B / ε ) by Theorem 3, again giving a contribution of size O ( B ) . Thisproves the claim. (cid:3) Remark.
We see that Proposition 4.3.4 for fixed d ≥ , and therefore alsoTheorems 1 and 4 for fixed degree, already follow from combining [7] with the resultsof [30] and Proposition 4.3.3. Similarly, for fixed degree d ≥ one can use the resultsof [26]. However, keeping track of the dependence on d in Section 3 permits us touse a considerably simpler argument for fixed d ≥ than in the works cited, andto furthermore obtain polynomial dependence on d .It remains to prove Theorems 1 and 4. This closely follows [7, Lemma 8, Theorem3]. The proofs are based on Proposition 4.3.4 and the following lemma.4.3.7. Lemma.
Let n ≥ and X ⊆ P n Q be a geometrically integral hypersurface ofdegree d . Then there exists a non-zero form F ∈ Z [ y , . . . , y n ] of degree at most ( n + 1)( d − such that F ( A ) = 0 whenever the hyperplane section H A ∩ X is notgeometrically integral, where A ∈ ( P n ) ∗ and H A ⊆ P n denotes the hyperplane cutout by the linear form associated with A .Proof. Suppose that X is given by f , a geometrically irreducible form of degree d .For A ∈ ( P n ) ∗ write A = ( a : a : . . . : a n ) ∈ ( P n ) ∗ . Assuming a = 0 , one has that H A ∩ X is not geometrically integral if and only if f ( − a a x − ... − a n a x n , x , . . . , x n ) HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 25 is reducible. Since n ≥ and since X is geometrically integral, we have for a genericchoice of B ∈ ( P n ) ∗ that H B ∩ X is also geometrically integral. Hence Theorem3.2.2 implies that there exists a non-zero form F in Z [ y , . . . , y n ] of degree at most d − such that F ( a , . . . , a n ) = 0 . Similarly, if a i = 0 , we produce a non-zeroform F i in Z [ y , . . . , y i − , y i +1 , . . . , y n ] such that F i ( a , . . . , a i − , a i +1 , . . . , a n ) = 0 . So F = Q ni =0 F i is as we want. (cid:3) Proof of Theorem 4.
Let n ≥ and X ⊆ A n Q be a geometrically integral hypersur-face of degree d ≥ described by a polynomial f ∈ Z [ x , . . . , x n ] with absolutelyirreducible highest degree part. We proceed by induction on n , where the base case n = 3 is Proposition 4.3.4.Now assume that n > and the theorem holds for all lower n . Let f d = h ( f ) be the homogeneous part of highest degree, which describes a hypersurface in P n − .By Lemma 4.3.7 and Corollary 4.1.2, there exists A = ( a , . . . , a n ) such that thehyperplane section { f d = 0 } ∩ { P a i x i = 0 } is geometrically integral of degree d ,with all a i having absolute value at most n ( d − .Now N aff ( f, B ) ≤ X | k |≤ n ( d − B N aff ( { f = 0 } ∩ { X a i x i = k } , B ) . For each k , the variety { f = 0 } ∩ { P a i x i = k } is a hypersurface in the affine plane { P a i x i = k } , which after a change of variables is described by a polynomial g ∈ Z [ x , . . . , x n − ] whose homogeneous part of highest degree is absolutely irreducibleby the construction of A . Now the induction hypothesis finishes the proof. (cid:3) Proof of Theorem 1.
We may assume that the variety in question is geometricallyirreducible by Remark 4.1.5, and can reduce to consideration of a hypersurface byProposition 4.3.2. Hence let n ≥ and consider an absolutely irreducible polynomial f ∈ Z [ x , . . . , x n ] homogeneous of degree d ≥ .Then f defines not only a projective hypersurface X in P n , but also an affinehypersurface in A n +1 , the cone of X . We now trivially have N ( f, B ) ≤ N aff ( f, B ) , so Theorem 4 finishes the proof. (cid:3) Remark.
Using the explicit exponents obtained in Proposition 4.3.4 and in theproof of Proposition 4.3.2 in Section 5, we can conservatively estimate e ( n ) ≤ n + 8 for the exponent in Theorem 4, and e ( n ) ≤ n for the exponent in Theorem 1.5. Reduction to hypersurfaces via projection
In this section we prove Propositions 4.3.1 and 4.3.2, which allowed us to reduceto the case of hypersurfaces in the proofs of our main theorems. This is an elabo-ration of familiar projection arguments, which classically show that every variety isbirational to a hypersurface, and which are used in the proofs of [7, Theorem 1] and[21, Theorem A]. The additional difficulty for us is that we have to keep track of thedependence on the degree of the variety throughout. Our main auxiliary result is:
Lemma.
Given a geometrically irreducible subvariety X ⊆ P n of dimension m < n − and degree d , one can find an ( n − m − -plane Λ disjoint from X and an ( m + 1) -plane Γ , both defined over Q , such that Λ ∩ Γ = ∅ , such that thecorresponding projection map p Λ , Γ : P n \ Λ → Γ satisfies (5.1) H ( p Λ , Γ ( P )) ≤ c n d n − m − H ( P ) for all P ∈ P n ( Q ) \ Λ , and such that p Λ , Γ | X is birational onto its image. Here c n isan explicit constant depending only on n . Because Λ is disjoint from X , the statement that p Λ , Γ | X is birational onto itsimage is equivalent to saying that p Λ , Γ ( X ) is again a variety of degree d , see [15,Example 18.16].In order to prove Lemma 5.1, we first concentrate on finding an appropriate Λ ,which we think of as living in the Grassmannian G ( n − m − , n ) consisting of all ( n − m − -planes in P n . It is well-known that the latter has the structure of an ( m + 2)( n − m − -dimensional irreducible projective variety through the Plückerembedding P n − m − ,n : G ( n − m − , n ) ֒ → P ν : Λ det( P , ..., P n − m − ) , where ν = (cid:0) n +1 n − m − (cid:1) − and ( P , ..., P n − m − ) is the ( n − m − × ( n + 1) matrixwhose rows are coordinates for n − m − independent points P i ∈ Λ . Here andthroughout this section, for a matrix M whose number of rows does not exceed itsnumber of columns, we write det( M ) to denote the tuple consisting of its maximalminors, with respect to some fixed ordering.Fixing such a Λ ∈ G ( n − m − and independent points P , ..., P n − m − ∈ Λ , wecan also consider the map π Λ : P n \ Λ → P µ : P det( P, P , ..., P n − m − ) , where µ = (cid:0) n +1 n − m (cid:1) − . Writing π Λ = ( π Λ , , ..., π Λ ,µ ) we see that the non-zero π Λ ,j ’scan be viewed as linear forms whose coefficients are coordinates of P n − m − ,n (Λ) ,modulo sign flips. Note that π Λ ,j ( P ) = 0 for all j if and only if P ∈ Λ . In particular π Λ is well-defined and easily seen to factor as(5.2) P n \ Λ p Λ , Γ −→ Γ ֒ → P µ for all ( m + 1) -planes Γ such that Γ ∩ Λ = ∅ .Another theoretical ingredient we need is the Chow point F X associated with anirreducible m -dimensional degree d variety X ⊆ P n . This is an irreducible multiho-mogeneous polynomial of multidegree ( d, d, . . . , d ) in m + 1 sets of n + 1 variablessuch that for all tuples ( H , H , . . . , H m +1 ) of m + 1 hyperplanes in P n one has F X ( H , . . . , H m +1 ) = 0 if and only if H ∩ H ∩ . . . ∩ H m +1 ∩ X = ∅ . See e.g. [14,Chapter 4].5.2. Lemma.
Let X be a geometrically irreducible degree d subvariety of P n havingdimension m < n − and consider G X = { Λ ∈ G ( n − m − , n ) | Λ ∩ X = ∅ and π Λ | X is birational onto its image } HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 27 with π Λ as above. This is a dense open subset of G ( n − m − , n ) whose complement,when viewed under the Plücker embedding, is cut out by hypersurfaces of degree lessthan ( m + 1) d .Proof. Given a hyperplane H ⊆ P µ we abusively write H ◦ π Λ for π − ( H ) ∪ Λ , sincethis is the hyperplane in P n cut out by the precomposition of π Λ with the linearform associated with H . Define a multihomogeneous degree ( d, d, . . . , d ) polynomial R X, Λ in m + 1 sets of µ + 1 variables by letting R X, Λ ( H , H , . . . , H m +1 ) = F X ( H ◦ π Λ , H ◦ π Λ , . . . , H m +1 ◦ π Λ ) . Note that its coefficients are degree ( m + 1) d polynomial expressions in the coordi-nates of P n − m − ,n (Λ) . We will show that(5.3) G X = { Λ ∈ G ( n − m − , n ) | R X, Λ is absolutely irreducible } , which implies that the complement of G X is precisely the vanishing locus of theNoether irreducibility polynomials from Theorem 3.2.2 evaluated in these coeffi-cients. This indeed yields expressions in the coordinates of P n − m − ,n (Λ) of degreeless than ( m + 1) d , where we note that not all these expressions can vanish iden-tically, since generic Λ ’s do not meet X and generic projections are known to bebirational [15, p. 224].We now prove (5.3). First note that Λ ∩ X = ∅ implies that R X, Λ vanishesidentically. Indeed, if P ∈ Λ then all hyperplanes of the form H ◦ π Λ pass through P , so if moreover P ∈ X we see that R X, Λ is identically zero. We can thereforeassume that Λ ∩ X = ∅ . This ensures that π Λ ( X ) is an irreducible projective varietyof dimension m , see [15, p. 134], so we can consider its Chow point F π Λ ( X ) , which isan irreducible multihomogeneous polynomial of multidegree (deg( π Λ ( X )) , deg( π Λ ( X )) , . . . , deg( π Λ ( X ))) in the same m +1 sets of µ +1 variables as in the case of R X, Λ . It has the property thatfor all tuples ( H , . . . , H m +1 ) of hyperplanes in P µ we have F π Λ ( X ) ( H , . . . , H m +1 ) = 0 if and only if H ∩ . . . ∩ H m +1 ∩ π Λ ( X ) = ∅ . But in this case π − ( H ) ∩ . . . ∩ π − ( H m +1 ) ∩ X = ∅ so that R X, Λ ( H , . . . , H m +1 ) = 0 . Conversely, if R X, Λ ( H , . . . , H m +1 ) = 0 thenthere exists a point P ∈ H ◦ π Λ ∩ . . . ∩ H m +1 ◦ π Λ ∩ X , which since Λ ∩ X = ∅ impliesthat π Λ ( P ) ∈ H ∩ . . . ∩ H m +1 ∩ π Λ ( X ) and hence that F π Λ ( X ) ( H , . . . , H m +1 ) = 0 .We conclude that F π Λ ( X ) and R X, Λ have the same vanishing locus and because theformer polynomial is irreducible there must exist some r ≥ such that R X, Λ = F rπ Λ ( X ) . In particular R X, Λ is irreducible if and only if r = 1 . But this is true if and only if π Λ ( X ) has degree d , which as we know holds if and only if π Λ | X is birational ontoits image. (cid:3) Lemma.
Using the assumptions and notation from Lemma 5.2, there exists an ( n − m − -plane Λ ∈ G X ( Q ) such that H (Λ) ≤ (( m + 1) d ) n − m − ( n − m − when considered under the Plücker embedding.Proof. Consider the rational map π : ( P n ) n − m − P ν : ( P , . . . , P n − m − ) det( P , . . . , P n − m − ) which is well-defined on the open U consisting of tuples of independent points.Observe that π ( U ) = G ( n − m − , n ) . By Lemma 5.2 there exists a polynomial F of degree less than ( m + 1) d which vanishes on the complement of G X but whichdoes not vanish identically on G ( n − m − , n ) . The polynomial Q := F det x x . . . x n x x . . . x n ... ... . . . ... x n − m − , x n − m − , . . . x n − m − ,n is multihomogeneous of multidegree (deg( F ) , . . . , deg( F )) in the n − m − blocks of n +1 variables corresponding to the rows of the displayed matrix. Clearly Q vanisheson the complement of U , while it is not identically zero because Q ( P , . . . , P n − m − ) = F ( π ( P , . . . , P n − m − )) for any tuple of independent points P i .Write Q = X j Q j ( x , . . . , x n ) R j ( x , . . . , x n − m − ,n ) for non-zero Q j and linearly independent polynomials R j . Lemma 4.1.1 helps us tofind a point P ∈ P n ( Q ) of height at most deg( F ) such that Q ( P ) = 0 . By the linearindependence of the R j ’s one sees that Q ( P , x , . . . , x n − m − ,n ) is not identicallyzero. Repeating the argument eventually yields a tuple of points P , P , . . . , P n − m − of height at most deg( F ) such that Q ( P , . . . , P n − m − ) = 0 . In particular this tuple ofpoints belongs to U , i.e., they are independent, and π ( P , P , . . . , P n − m − ) ∈ G X ( Q ) .From this the lemma follows easily. (cid:3) Proof of Lemma 5.1.
Let Λ be the Q -rational ( n − m − -plane produced by theproof of Lemma 5.3. In particular Λ ∩ X = ∅ and π Λ | X is birational onto its image.Then for all ( m + 1) -planes Γ such that Γ ∩ Λ = ∅ the projection map p Λ , Γ | X is alsobirational onto its image, thanks to the factorization from (5.2).The proof of Lemma 5.3 moreover shows that Λ can be assumed to be the linearspan of rational points P , . . . , P n − m − ∈ P n satisfying H ( P i ) ≤ ( m + 1) d =: B . By Lemma 5.4 below we can find linear forms L , L , . . . , L n − m − with integralcoefficients whose absolute value is bounded by B := p ( n − m − n + 1) B n − m − such that L i vanishes on P , . . . , P i − , P i +1 , . . . , P n − m − but not on P i . Togetherthese linear forms cut out an ( m + 1) -plane Γ such that Γ ∩ Λ = ∅ . Furthermore(5.4) p Λ , Γ ( P ) = P − L ( P ) L ( P ) P − . . . − L n − m − ( P ) L n − m − ( P n − m − ) P n − m − HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 29 for all P ∈ P n \ Λ . So we have(5.5) H ( p Λ , Γ ( P )) ≤ ( n − m )(( n + 1) B B ) n − m − H ( P ) = cd n − m − H ( P ) for some constant c that is easily bounded by an expression purely in n . (cid:3) Lemma.
Let
B, r, s ∈ Z ≥ be integers such that s < r . Consider a linear systemof linearly independent equations P rk =1 a ik x k = 0 for i = 1 , . . . , s , where all a ij areintegers satisfying | a ij | ≤ B . There exists a non-zero tuple of integers ( x , x , . . . , x r ) violating the first equation but satisfying all other equations such that (5.6) | x i | ≤ p ( s − rB s − for all i .Proof. This follows from [4, Theorem 2], which strengthens Theorem 3.3.4. It en-sures the existence of r − s + 1 linearly independent tuples of integers ( x , x , . . . , x r ) satisfying the last s − equations and meeting the bound (5.6). Since the space ofsolutions to the full linear system of s equations has dimension r − s , at least oneof these tuples must violate the first equation. (cid:3) We can now prove Propositions 4.3.1 and 4.3.2, reducing the situation of a generalvariety to a hypersurface.
Proof of Proposition 4.3.2.
Let X be a geometrically integral projective variety in P n of dimension m and degree d , where we may assume that n > m + 1 . We considera projection p Λ , Γ as in Lemma 5.1. By dropping appropriately chosen coordinates,its image X ′ can be viewed as a hypersurface in P m +1 , birational to X and hencealso of degree d . In each fibre of p Λ , Γ there are at most d points. The height relationfrom Lemma 5.1 now immediately implies N ( X, B ) ≤ dN ( X ′ , c n d n − m − B ) for all B ≥ . This proves the claim for m > . For m = 1 , consider the normaliza-tion ˜ X → X and compose it with the morphism X → X ′ induced by p Λ , Γ to finda resolution of singularities ˜ X → X ′ . The latter map is one-to-one away from thesingular points of X ′ , which together have no more than ( d − d − preimagesby [19, Theorem 17.7(b)]. But then the same claims must apply to X → X ′ , yieldingthe stronger bound N ( X, B ) ≤ N ( X ′ , c n d n − B ) + d , as wanted. (cid:3) Proof of Proposition 4.3.1.
Let X be a geometrically integral affine variety in A n ofdimension m and degree d , where we may assume that m < n − . Let Z be theprojective closure of X in P n ; we apply Lemma 5.1 and shall argue later that we cantake the ( n − m − -plane Λ to be contained in the hyperplane P n − at infinity. Let Z ′ ⊆ Γ be the image of Z under the projection p Λ , Γ . As above, by dropping somecoordinates we can view Γ as P m +1 = A m +1 ⊔ P m where p Λ , Γ ( P n − \ Λ) correspondsto P m . In particular p Λ , Γ maps X to the affine part X ′ = Z ′ ∩ A m +1 of Z ′ . Consider P , P , . . . , P n − m − and L , L , . . . , L n − m − as in the proof of Lemma 5.1.Let P ∈ X be a point having integer coordinates; when considered as a projectivepoint of Z its coordinate at infinity is . Since the coordinates at infinity of the P i ’s are , the projection formula (5.4) shows that p Λ , Γ ( P ) ∈ Z ′ admits integercoordinates such that the coordinate at infinity is L ( P ) L ( P ) · · · L n − m − ( P n − m − ) , regardless of the choice of P . As a consequence, this is a multiple of the denominatorsappearing among the coordinates of p Λ , Γ ( P ) when viewed as an affine rational pointof X ′ . Therefore, postcomposing with a coordinate scaling map A m +1 → A m +1 , weobtain another variety X ′ in A m +1 such that every integral point P of X is mappedto an integral point of X ′ whose height satisfies the same upper bound as in (5.5).All fibres of this map X → X ′ have at most d points, and in the case of curves themap is even one-to-one away from the singular points on X ′ . So we can conclude asin the proof of Proposition 4.3.2.It remains to argue why we can take Λ in the hyperplane at infinity. We firstclaim that the “good set” G Z from Lemma 5.2 has a non-empty intersection withthe Grassmannian parametrizing ( n − m − -planes Λ contained in P n − . Indeed,it is apparent that the generic such Λ does not intersect the ( m − -dimensionalset Z ∩ P n − and hence satisfies Λ ∩ Z = ∅ . Furthermore, the argument from [15,p. 224] showing that generic projections are birational leaves enough freedom todraw the same conclusion when restricting to projections from planes at infinity.More precisely, if m = n − then it suffices to project from a point outside thecone spanned by Z and some random point q ∈ Z . Since this cone is irreducible ofdimension at most m + 1 = n − and since Z P n − , the generic point at infinityindeed meets this requirement. If m < n − then the desired conclusion follows byapplying the foregoing argument to n − m − successive projections from points.So we can redo the proof of Lemma 5.3 starting from a polynomial F of degreeless than ( m + 1) d which vanishes on the complement of G X but which does notvanish identically on the Grassmannian of ( n − m − -planes that are containedin the hyperplane at infinity; we just argued that such an F exists. Then one canproceed with the same polynomial Q as before, but with zeroes substituted for thevariables x , x , . . . , x n − m − , . (cid:3) Lower bounds
We conclude with some lower bounds showing that one cannot make the depen-dence on d sub-polynomial. Our main auxiliary tool is the following lemma.6.1. Lemma.
For each pair of integers d ≥ , n ≥ there exists an absolutelyirreducible degree d polynomial f ∈ Q [ x , x , . . . , x n ] which vanishes at all integralpoints ( r , r , . . . , r n ) for which | r i | ≤ ⌊ ( d − / n ⌋ for all i . HE DIMENSION GROWTH CONJECTURE, POLYNOMIAL IN THE DEGREE 31
Proof.
The lemma is immediate if d = 1 , so we can assume that d ≥ . We claimthat there exists a polynomial x d + x d + . . . + x dn − + x d − n + X ≤ i ,...,i n ≤⌊ ( d − /n ⌋ a i ,...,i n x i x i · · · x i n n which vanishes simultaneously at the integral points ( r , r , . . . , r n ) satisfying ⌊ ( d − / n ⌋ − ⌊ ( d − /n ⌋ ≤ r i ≤ ⌊ ( d − / n ⌋ for all i . From this the lemma follows, because indeed ⌊ ( d − / n ⌋ − ⌊ ( d − /n ⌋ ≤−⌊ ( d − / n ⌋ and because the polynomial is absolutely irreducible, as its Newtonpolytope is indecomposable; see e.g. [12, Theorem 4.11]. To verify the claim, notethat every point ( r , r , . . . , r n ) imposes a linear condition on the coefficients a i ,...,i n ,together resulting in a linear system of ( ⌊ ( d − /n ⌋ + 1) n equations in the samenumber of unknowns. It suffices to see that the matrix corresponding to its linearpart is non-singular. But this matrix is the n th Kronecker power of the Vandermondematrix ( r i ) r,i where r and i range over {⌊ ( d − / n ⌋ − ⌊ ( d − /n ⌋ , . . . , ⌊ ( d − / n ⌋} resp. { , . . . , ⌊ ( d − /n ⌋} . Therefore its determinant is a power of the determinant of this Vandermonde matrix,from which the desired conclusion follows. (cid:3)
Proof of Proposition 5. If d = 1 , then we let X be a line resp. conic through acoordinate point, so that we can take B = 1 . If d ≥ then we consider the affinecurve defined by the polynomial f from the proof of the foregoing lemma for n = 2 .Let X be its projective closure, which has an extra height point at infinity. With B = ⌊ ( d − / ⌋ − ⌊ ( d − / ⌋ one observes that N ( X, B ) ≥ ( ⌊ ( d − / ⌋ + 1) + 1 ≥ d / d / · / ≥ d / · B /d . (cid:3) Note that using the same f and B one also finds that N aff ( f, B ) ≥ ( ⌊ ( d − / ⌋ + 1) ≥ d d B /d log B for all d ≥ , confirming our claim that, in the statement of Theorem 3, it is impos-sible to replace the quartic dependence on d by any expression which is o ( d / log d ) .In arbitrary dimension, the same reasoning shows that there exists a positive con-stant c = c ( n ) such that for all integers d > we can find an absolutely irreducibledegree d polynomial f ∈ Q [ x , x , . . . , x n ] along with an integer B ≥ such that N aff ( f, B ) ≥ cd B n − and N ( X, B ) ≥ cdB dim X , where X ⊆ P n Q denotes the integral degree d hypersurface defined by the homog-enization of f . This shows that Theorems 1 and 4 cannot hold with e < resp. e < . References
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