An Incidence Theorem in Higher Dimensions
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Abstract
We prove almost tight bounds on the number of incidences between points and kdimensional varieties of bounded degree in R ^{ d }. Our main tools are the polynomial ham sandwich theorem and induction on both the dimension and the number of points.
Keywords
Szemerédi–Trotter type incidence theorems Sumproduct bounds1 Introduction
Given a collection P of points in some space, and a collection L of sets in that same space, let I(P,L):={(p,ℓ)∈P×L:p∈ℓ} be the set of incidences. One of the objectives in combinatorial incidence geometry is to obtain good bounds on the cardinality I(P,L) on the number of incidences between finite collections P,L, subject to various hypotheses on P and L. For instance, we have the classical result of Szemerédi and Trotter [48]:
Theorem 1.1
(Szemerédi–Trotter Theorem [48])
This theorem is usually stated in the twodimensional setting d=2, but the higherdimensional case is an immediate consequence by applying a generic projection from R ^{ d } to R ^{2} (see Sect. 5.1 for a discussion of this argument). It is known that this bound is sharp except for the constant C; see [48].
Various mathematical problems can be transformed to a question about incidence bounds of Szemerédi–Trottertype. For instance, Elekes [11] used the above theorem in an unexpected fashion to obtain new bounds on the sumproduct problem. In the 1990s, Wolff [53] observed that bounds on the number of incidences might be used in problems related to the Kakeya conjecture, one of the central conjectures in harmonic analysis. Bennett, Carbery and Tao [6] established a connection between multilinear Kakeya estimates and bounds on number of incidences between points and lines in three dimensions. Very recently, Guth and Katz [18] used bounds on the number of incidences between points and lines in three dimensions as part of their solution to the Erdős Distinct Distances Problem. They used an important tool, the socalled polynomial ham sandwich theorem. This theorem will be a crucial part of this paper as well. The applicability of the polynomial ham sandwich theorem to Szemerédi–Trottertype theorems was also recently emphasized in [25].
Further applications of Szemerédi–Trottertype incidence bounds in mathematics and theoretical computer science, as well as several open problems, are discussed in the surveys and books of Elekes [12], Székely [47], Pach and Sharir [36], Brass, Moser, and Pach [7], and Matoušek [29].
In the survey [12], Elekes listed some nice applications of pointline incidence bounds in the complex plane ℂ^{2}, where the lines are now complex lines (and thus are also real planes). In this paper he referred to a (then) recent result of Tóth [52] which proves the pointline incidence bound (1.1) in this situation (with a different constant C). Up to this constant multiplier, this bound is optimal. Our argument is different from and simpler than the one in Tóth’s paper; however, the bounds in this paper are slightly weaker than those in [52].
The main goal of our paper is to establish nearsharp Szemerédi–Trottertype bounds on the number of incidences between points and kdimensional algebraic varieties in R ^{ d } for various values of k and d, under some “pseudoline” hypotheses on the algebraic varieties; see Theorem 2.1 for a precise statement. In particular, we obtain nearsharp bounds for pointline incidences in ℂ^{2}, obtaining a “cheap” version of the result of Tóth mentioned previously.
Our argument is based on the “polynomial method” as used by Guth and Katz [18], combined with an induction on the size of the point set P. The inductive nature of our arguments causes us to lose an arbitrarily small epsilon term in the exponents, but the bounds are otherwise sharp.
As in [18], our arguments rely on an efficient cell decomposition provided to us by the polynomial ham sandwich theorem (see Corollary 5.3). However, the key innovation here, as compared to the arguments in [18], is that this decomposition will only be used to partition the point set into a bounded number of cells, rather than a large number of cells. (Similar recursive space partitioning techniques were used by Agarwal and Sharir [1].) This makes the contribution of the cell boundaries much easier to handle (as they come from varieties of bounded degree, rather than large degree). The price one pays for using this milder cell decomposition is that the contribution of the cell interiors can no longer be handled by “trivial” bounds. However, it turns out that one can use a bound coming from an induction hypothesis as a substitute for the trivial bounds, so long as one is willing to concede an epsilon factor in the inductive bound. This technique appears to be quite general, and suggests that one can use induction to significantly reduce the need for quantitative control of the geometry of highdegree algebraic varieties when applying the polynomial method to incidence problems, provided that one is willing to lose some epsilons in the final bounds.
Our results have some similarities with existing results in the literature; we discuss these connections in Sect. 2.2.
Notation
We use the usual asymptotic notation X=O(Y) or X≪Y to denote the estimate X≤CY for some absolute constant C. If we need the implied constant C to depend on additional parameters, we indicate this by subscripts, thus for instance X=O _{ d }(Y) or X≪_{ d } Y denotes the estimate X≤C _{ d } Y for some quantity C _{ d } depending on d.
2 Main Theorem
In what follows we are going to use some standard notations and definitions from algebraic geometry. In Sect. 4 we provide the basic definitions and tools we will need from algebraic geometry, although this is only the barest of introductions and we refer the reader to standard textbooks like [16, 20, 21, 32] (or general reference works such as [15, 22]) for a detailed treatment.
Our main result (proven in Sect. 5) is as follows.
Theorem 2.1
(Main Theorem)
 (i)
For each ℓ∈L, ℓ is a real algebraic variety, which is the restriction to R ^{ d } of a complex algebraic variety ℓ _{ℂ} of dimension k and degree at most C _{0}.
 (ii)
If ℓ,ℓ′∈L are distinct, then there are at most C _{0} points p in P such that \((p,\ell), (p,\ell') \in{\mathcal{I}}\).
 (iii)
If p,p′∈P are distinct, then there are at most C _{0} varieties ℓ in L such that \((p,\ell), (p',\ell) \in {\mathcal{I}}\). (Note that for C _{0}=1, this is equivalent to (ii).)
 (iv)
If \((p,\ell) \in{\mathcal{I}}\), then p is a smooth (real) point of ℓ, with a real tangent space. In other words, for each \((p,\ell)\in{\mathcal{I}}\), there is a unique tangent space T _{ p } ℓ of ℓ at p, which is a kdimensional real affine space containing p.
 (v)
If ℓ,ℓ′∈L are distinct, and p∈P are such that \((p,\ell), (p,\ell') \in{\mathcal{I}}\), then the tangent spaces T _{ p } ℓ and T _{ p } ℓ′ are transverse, in the sense that they only intersect at p.
Remark 2.2
The condition d≥2k is natural, as we expect the tangent spaces T _{ p } ℓ, T _{ p } ℓ′ in Axiom (v) to be kdimensional;^{1} if d<2k, such spaces cannot be transverse in R ^{ d }. As we will see shortly, the most interesting applications occur when d=2k and k≥1, with C _{0} being an extremely explicit constant such as 1, 2, or 4, the varieties in L being smooth (e.g. lines, planes, or circles), and the incidences \({\mathcal{I}}\) comprising all of I(P,L); but for inductive reasons it is convenient to consider the more general possibilities for d, k, C _{0}, L, and \({\mathcal{I}}\) allowed by the above theorem. The constants \(\frac{3}{2}\) could easily be replaced in the argument by any constant greater than 1. We choose the constant here to be less than 2 so that the bound (2.1) can be used to control rrich points and lines for r as low as 2 (though in that particular case, the trivial bounds in Lemma 5.1 already suffice).
In Sect. 3 we will sketch a simplified special case of the above theorem which can be proven using less of the machinery from algebraic geometry; it is conceivable that many of our applications can be handled by this simpler method. On the other hand we expect that there are further applications where the whole generality of our result is needed.
2.1 Applications
Suppose we specialize Theorem 1.1 to the case when the varieties in L are kdimensional affine subspaces, such that any two of these subspaces meet in at most one point. Then one easily verifies that Axioms (i)–(v) hold with C _{0}=1 and \({\mathcal{I}}:=I(P,L)\). We conclude:
Corollary 2.3
(Cheap Szemerédi–Trotter for kFlats)
Except for the ε loss, this answers a conjecture of Tóth [52, Conjecture 3] affirmatively. The hypothesis d≥2k can be dropped for the trivial reason that it is no longer possible for the kdimensional subspaces in L to intersect each other transversely for d<2k, but of course the result is not interesting in this regime.
As special cases of Corollary 2.3, we almost recover (but for epsilon losses) the classic Szemerédi–Trotter theorem (Theorem 1.1), as well as the complex Szemerédi–Trotter theorem of Tóth [52]. More precisely, from the k=2 case of Corollary 2.3, we have
Corollary 2.4
(Cheap Complex Szemerédi–Trotter)
We will sketch a separate proof of this corollary in Sect. 3, in order to motivate the more complicated argument needed to establish Theorem 1.1 in full generality.
One can also establish the same bound for the quaternions ℍ. Define a quarternionic line to be any set in ℍ^{2} of the form {(a,b)+t(c,d):t∈ℍ} for some a,b,c,d∈ℍ with (c,d)≠(0,0). Because ℍ is a division ring, we see that any two distinct quarternionic lines meet in at most one point. After identifying ℍ with R ^{4}, we conclude:
Corollary 2.5
(Cheap Quaternionic Szemerédi–Trotter)
We can also replace lines with circles. Define a complex unit circle to be a set of the form {(z,w)∈ℂ^{2}:(zz _{0})^{2}+(ww _{0})^{2}=1} for some z _{0},w _{0}∈ℂ. It is easy to verify that complex unit circles are real algebraic varieties in ℂ^{2}≡R ^{4} of (real) dimension 2 and (real) degree 2×2=4 (because the real and imaginary parts of the defining equation (zz _{0})^{2}+(ww _{0})^{2}=1 are both quadratic constraints), that two complex unit circles meet in at most two points, and that two points determine at most two complex unit circles. It is possible for a point to be incident to two distinct complex unit circles in such a fashion that their tangent spaces (which are complex lines, or real planes) coincide, when the two circles are reflections of each other across their common tangent space; however, if we first pigeonhole the incidences into O(1) classes, based on the orientation of the radial vector (zz _{0},ww _{0}) connecting the center (z _{0},w _{0}) of the complex unit circle to the point (z,w), then we can eliminate these unwanted tangencies. We conclude
Corollary 2.6
(Cheap Szemerédi–Trotter for Complex Unit Circles)
This gives the following application to the complex unit distance problem:
Corollary 2.7
(Complex Unit Distances)
Indeed, this claim follows from applying (2.5) to the family L of complex unit circles with centers in P. We remark that the realvariable analogue of this result was established by Spencer, Szemerédi, and Trotter [44].
Corollary 2.8
(SumProduct)
Proof
We will apply Corollary 2.3, where the kflats are given by \(\{ (\vec{x}, \vec{y}) \in {\mathbf{R}}^{k} \times{\mathbf{R}}^{k}: \vec {y}=A(\vec{x}\vec {v})\}\) with \(\vec{v}\in V\) and \(A\in\mathcal{A}\) and the points are the elements of the Cartesian product \(\{V+W\}\times\{{\mathcal{A}}W\}\). Any two flats have at most one common point since det(AB)≠0 and each of them has dimension k in the 2kdimensional real space. Any point with coordinates \((\vec{w}+\vec{v}, A\vec{w})\) is incident to \(\vec{y}=A(\vec{x}\vec{v})\). So, we have n ^{2} kdimensional flats and \(V+W{\mathcal{A}}W\) points where each flat is incident to at least n points. We can apply Corollary 2.3 now to prove our bound. □
Remark 2.9
… Given a group G of transformations of R ^{ d } and a finite pointset \(\mathcal{P}\subset{\mathbf{R}}^{d}\) we shall be interested in the number of transformations φ∈G which map many points of \(\mathcal {P}\) to some other points of \(\mathcal{P}\) …
Here we consider affine transformations in R ^{ d }. We need some notation. An affine transformation is rrich with respect to \(\mathcal {P}\) if \(A(\mathcal{P})\cap\mathcal{P}\geq r\). Elekes’ question is to bound the number of rrich transformations. A finite set of affine transformations, \(\mathcal{A}\) is said to be pairwise independent if A ^{1} B has at most one fixpoint for any \(A,B\in\mathcal{A}\).
Corollary 2.10
(Affine Transformations)
Given an nelement pointset \(\mathcal{P}\subset{\mathbf{R}}^{d}\), and let ε>0 and r≥2. Any set X of pairwise independent rrich affine transformations has cardinality at most An ^{4+ε }/r ^{3}, where A=A _{ ε,d }>0 depends only on ε and d.
Proof
Each affine transformation in X can be written as \(\vec {x}\rightarrow A\vec{x}+\vec{v}\), which we can view as a dflat \(\{(\vec{x}, \vec{y}) \in{\mathbf{R}}^{d} \times{\mathbf{R}}^{d}: \vec{y} = A\vec{x} + \vec{v}\}\) in R ^{ d }×R ^{ d }. Each such flat is incident to at least r points of the Cartesian product \(\mathcal{P}\times\mathcal{P}\). There are X ddimensional flats and n ^{2} points where each flat is incident to at least r points. Any two flats have at most one common point since the transformations are pairwise independent (two or more common points would mean that the corresponding affine transformations are identical on a line). One can apply Corollary 2.3 again to prove the bound X≪n ^{4+ε }/r ^{3}. □
2.2 Comparison with Existing Results
The polynomial partitioning method is not the only method to establish incidence bounds between points and varieties. In particular, there are other methods to obtain cell decompositions which can achieve a similar effect to the decomposition given by the polynomial Ham Sandwich theorem, though the hypotheses on the configuration of points and varieties can be quite different from those considered here. A model case is when the point set is assumed to be homogeneous, which roughly speaking means that the point set resembles a perturbation of a grid. In such cases one can use the cubes of the grid to form the cells. For instance, in [42] and [43] sharp incidence bounds between a homogeneous set of points and kdimensional subspaces were given. In [27] sharp pointpseudoplane incidence bounds were proved in R ^{3}. Similar bounds on pointsurface incidences were proved for the nonhomogeneous case by Zahl [54].
Elekes [13], Sharir and Welzl [40], and Guth and Katz [17] gave bounds on the number of joints. We omit the details but we should mention that the latter paper in particular uses the polynomial space partition method to give a satisfactory bound on the number of joints.
Pach and Sharir [34, 35] considered incidences between points and pseudolines in the plane—curves which obey axioms similar to Axioms (ii) and (iii) in Theorem 2.1, using crossing number inequalities. Such methods work particularly well in the plane, but are somewhat difficult to extend to higher dimensions; for some partial results in three and higher dimensions, see [3].
In the hyperplane case k=d1, sharp incidence bounds were obtained in [14] (of course, the transversality hypothesis needs to be modified in this regime).
In [24], Fourieranalytic methods were used to obtain incidence bounds of Szemerédi–Trottertype. In this method, the manifolds ℓ are not required to be algebraic varieties, but they are required to obey certain regularity hypotheses relating to the smoothing properties an associated generalized Radon transform (which usually forces them to be fairly high dimensional. Also, the point set P is assumed to obey a homogeneity assumption.
In [41] a simple proof was given to a Szemerédi–Trottertype bound for incidences between complex points and lines, however, the point set is assumed to be of Cartesian product of the form A×B⊂ℂ^{ d }.
3 A Special Case
Before we prove Theorem 2.1, we illustrate the key elements of the proof by sketching the proof of the (cheap) complex Szemerédi–Trotter Theorem which was stated earlier as Corollary 2.4. In this lowdimensional setting one can avoid an induction on dimension, instead using the crossing number machinery of Székely [46] to deal with the contribution of various lower dimensional objects. The reader who is impatient to get to the proof of the full theorem may skip this section if desired.
One can suppose that every point has at least half of the average incident lines w.l.o.g. (If a point has less than \({\mathcal{I}}/2m\) lines incident to it then simply remove it from P.) Then counting the points gives a lower bound on the incidences, e.g. if a region of ℂ^{2} contains k points then there are at least \(k{\mathcal{I}}/2m\) incidences between the k points and the set L.
If most of the incidences are in R ^{4}∖{Q=0} (inside a cell) then a simple doublecounting argument gives the desired bound. This is the case when we are going to use the induction hypothesis. If most of the incidences are on {Q=0} then we will bound the number of incidences directly.
So either \({\mathcal{I}}_{i}\leq mD^{2}/2 + 4P_{i}\) or \({\mathcal {I}}_{i}\leq 4(mDP_{i})^{2/3}\). Summing the number of incidences over the P _{ i }, i=1,…,D gives the desired inequality (3.3) (noting that the implied constants can depend on M and hence on D).
Remark 3.1
One can also view this argument from a recursive perspective rather than an inductive one. With this perspective, one starts with a collection of n points and m lines and repeatedly passes to smaller configurations of about n/M points and a smaller number of lines as well. Thus, one expects about log_{ M } n iterations in procedure. Collecting all the bounds together to obtain a final bound of the form An ^{2/3} m ^{2/3} (ignoring the lower order terms n,m for now), we see that with each iteration, the constant A increases by a bounded multiplicative factor (independent of M), assuming that A was chosen sufficiently large depending on M. Putting together these increases, one obtains a final value of A of the shape \(C_{M} \times C^{\log_{M} n}\); letting M become large, this gives bounds of the shape C _{ ε } n ^{ ε } as claimed. The key point is that the main term in the estimate only grows by a constant factor independent of M with each step of the iteration; the lower order terms, on the other hand, are permitted to grow by constants depending on C _{ M }, as they can be absorbed into the main term (using the reduction to the regime (3.2)).
4 Some Algebraic Geometry
In this section we review some notation and facts from algebraic geometry that we will need here. Standard references for this material include [16, 20] or [32].
It will be convenient to define algebraic geometric notions over the field ℂ, as it is algebraically complete. However, for our applications we will only need to deal with the real points of algebraic sets.
Definition 4.1
(Algebraic Sets)
The intersection of any subset of ℂ^{ d } with R ^{ d } will be referred to as the real points of that subset. A real algebraic variety is the real points V _{ R } of a complex algebraic variety V.^{3}
It is known (see e.g. [16, Sect. 1.3]) that to any rdimensional variety one can associate a unique natural number D, called the degree of V, with the property that almost every codimension r affine subspace of ℂ^{ d } intersects V in exactly D points. Thus, for instance, if P:ℂ^{ d }→ℂ is an irreducible polynomial of degree D, then the hypersurface {P=0} has dimension d1 (and thus codimension 1) and degree D.
We do not attempt to define the notions of degree and dimension directly for real algebraic varieties, as there are some subtle issues that arise in this setting (see e.g. [37]). However, in our applications every real algebraic variety will be associated with a complex one, which of course will carry a notion of degree and dimension. Later on (by using Proposition 4.6 below) we will see that we may easily reduce to the model case in which the real algebraic varieties have full dimension inside their complex counterparts, in the sense that the real tangent spaces have the same dimension as the complex ones.
Every algebraic set can be uniquely decomposed as the union of finitely many varieties, none of which are contained in any other (see e.g. [32, Proposition I.5.3]). We define the dimension of the algebraic set to be the largest dimension of any of its component varieties.
If V is an rdimensional variety in ℂ^{ d }, and P:ℂ^{ d }→V is a polynomial which is not identically zero on V, then every component of V∩{P=0} has dimension r1 (see [32, Sect. I.8]).
A basic fact is that the degree of a variety controls its complexity:^{4}
Lemma 4.2
(Degree Controls Complexity)
Proof
See [8, Theorem A.3] or [31]. Indeed, one can take P _{1},…,P _{ m } to be a linear basis for the vector space of all the polynomials of degree at most D that vanish identically on V. □
We have the following converse:
Lemma 4.3
(Complexity Controls Degree)
A smooth point of a kdimensional algebraic variety V is an element p of V such that V can be locally described by a smooth kdimensional complex manifold in a neighbourhood of p. Points in V that are not smooth will be called singular. We let \(V^{\operatorname{smooth}}\) denote the smooth points of V, and \(V^{\operatorname{sing}} := V \backslash V^{\operatorname{smooth}}\) denote the singular points.
It is well known that “most” points in an algebraic variety V are smooth (see e.g. [50, Theorem 5.6.8]). In fact, we have the following quantitative statement:
Proposition 4.4
(Most Points Smooth)
Let V be a kdimensional algebraic variety in ℂ^{ d } of degree at most D. Then \(V^{\operatorname{sing}}\) can be covered by O _{ D,d }(1) algebraic varieties in V of dimension at most k1 and degree O _{ D,d }(1).
Proof
We will prove the more general claim that for any integer r with k≤r≤d, there exists an algebraic variety W in ℂ^{ d } containing V of dimension at most r and degree O _{ D,d }(1), such that \(V \cap W^{\operatorname{sing}}\) can be covered by O _{ D,d }(1) algebraic varieties of dimension at most k1 and degree O _{ D,d }(1). If we apply this claim with r=k, then W must equal V, giving the proposition (after restricting the algebraic varieties produced by the claim to V).
We establish the claim by downward induction on r. The case r=d is trivial, as we can set W=ℂ^{ d }, so assume now that k≤r<d and that the claim has already been established for r+1. Thus there exists an algebraic variety W containing V of dimension at most r+1 and degree O _{ D,d }(1) such that \(V \cap W^{\operatorname {sing}}\) can be covered by O _{ D,d }(1) algebraic varieties of dimension at most k1 and degree O _{ D,d }(1).
 (i)
The gradient ∇P _{1} does not vanish identically on V.
 (ii)
∇P _{1} vanishes identically on V, but does not vanish identically on W.
 (iii)
∇P _{1} vanishes identically on W.
Suppose we are in case (i). Let W′:={x∈W:P _{1}(x)=0}. This algebraic set is strictly contained in W and thus has dimension precisely r, by the discussion prior to Lemma 4.2. By Lemma 4.2, W′ is cut out by O _{ D,d }(1) polynomials of degree O _{ D,d }(1). By Lemma 4.3, W′ is thus the union of O _{ D,d }(1) varieties of dimension r and degree O _{ D,d }(1). One of these varieties, call it W″, must contain V.
Let x be a point in \(V \cap W^{\operatorname{smooth}}\) with ∇P _{1}(x)≠0, then by the (complex)^{5} inverse function theorem, the set {x∈W:P _{1}(x)=0} is a smooth rdimensional complex manifold in a neighbourhood of x. As such, we see that x is a smooth point of W″. On the other hand, by Lemma 4.3, the set {x∈V:∇P _{1}(x)=0} can be covered by O _{ D,d }(1) varieties of dimension at most k1 and degree O _{ D,d }(1). We thus see that the claim holds for r by using W″ in place of W.
Now suppose we are in case (ii). Then there is a partial derivative \(\partial_{x_{i}} P_{1}\) of P _{1} that vanishes identically on V but does not vanish identically on W. We then replace P _{1} by \(\partial_{x_{i}}P_{1}\) (lowering the degree by one) and return to the above subdivision of cases. We continue doing this until we end up in case (i) or case (iii). Since case (ii) cannot hold for polynomials of degree zero or one, we see that we must eventually leave case (ii) and end up in one of the other cases.
Finally, suppose we are in case (iii). Then, by the fundamental theorem of calculus, P _{1} is constant in a neighbourhood of every smooth point of W. In particular, there is some constant c for which set {P _{1}=c} is nonempty and has the same dimension as W, and thus (by irreducibility of W) we conclude that P _{1} is constant on W; since P _{1} vanishes on V, it thus vanishes on W, a contradiction. Thus case (iii) cannot actually occur, and we are done. □
We may iterate this proposition (performing an induction on the dimension k) to obtain
Corollary 4.5
(Decomposition into Smooth Points)
Let V be a kdimensional algebraic variety in ℂ^{ d } of degree at most D. Then one can cover V by \(V^{\operatorname{smooth}}\) and O _{ D,d }(1) sets of the form \(W^{\operatorname{smooth}}\), where each W is an algebraic variety in V of dimension at most k1 and degree O _{ D,d }(1).
Finally, we address a technical point regarding the distinction between real and complex algebraic varieties. If p∈ℓ⊂R ^{ d } is a smooth real point of a kdimensional complex algebraic variety ℓ _{ℂ}∈ℂ^{ d }, then it must have a kdimensional complex tangent space T _{ p } ℓ _{ℂ}⊂ℂ^{ d }, by definition. However, its real tangent space T _{ p } ℓ:=T _{ p } ℓ _{ℂ}∩R ^{ d } may have dimension smaller than k. For instance, in the case k=1, d=2, the complex line ℓ _{ℂ}:={(z,w)∈ℂ^{2}:z=iw} has a smooth point at (0,0) with a onedimensional complex tangent space (also equal to ℓ _{ℂ}), but the real tangent space is only zerodimensional. Of course, in this case, the real portion ℓ:=ℓ _{ℂ}∩R ^{2} of the complex line is just a zerodimensional point. This phenomenon generalizes:
Proposition 4.6
 (i)
The real points V _{ R } of V are covered by the smooth points \(W^{\operatorname{smooth}}\) of O _{ D,d }(1) algebraic varieties W of dimension at most k1 and degree O _{ D,d }(1) which are contained in V.
 (ii)
For every smooth real point \(p \in V^{\operatorname {smooth}}_{\mathbf{R}}\) of V, the real tangent space T _{ p } V _{ R }:=T _{ p } V∩R ^{ d } is kdimensional (and thus T _{ p } V is the complexification of T _{ p } V _{ R }). In particular, \(V^{\operatorname {smooth}}_{\mathbf{R}}\) is kdimensional.
Proof
By Lemma 4.2, V can be cut out by polynomials P _{1},…,P _{ m } of degree O _{ D,d }(1) with m=O _{ D,d }(1). Let \(\tilde{V}\) be the algebraic set cut out by both P _{1},…,P _{ m } and their complex conjugates \(\overline{P_{1}},\ldots,\overline{P_{m}}\), defined as the complex polynomials whose coefficients are the conjugates of those for P _{1},…,P _{ m }. Clearly V and \(\tilde{V}\) have the same real points. If \(\tilde{V}\) is strictly smaller than V, then it has dimension at most k1, and by Lemma 4.3 and Corollary 4.5 we are thus in case (i). Now suppose instead that V is equal to \(\tilde{V}\). Then at every smooth point p of V (or \(\tilde{V}\)), the kdimensional complex tangent space \(T_{p} V = T_{p}\tilde{V}\) is cut out by the orthogonal complements of the gradients ∇P _{1},…,∇P _{ m } and their complex conjugates. As such, it is manifestly closed with respect to complex conjugation, and is thus the complexification of its real counterpart \(T_{p} \tilde{V}_{\mathbf{R}}\). We are thus in case (ii). □
Note that by iterating the above proposition, we may assume that each of the varieties W occurring in case (i) obey the properties stated in case (ii).
5 Proof of Main Theorem
We now prove Theorem 2.1. We will prove this theorem by an induction on the quantity d+k. The case d+k=0 is trivial, so we assume inductively that d+k≥1 and that the claim has already been proven for all smaller values of d+k.
We can dispose of the easy case k=0, because in this case L consists entirely of points, and in particular each ℓ in L is incident to at most one point in P, giving the bound \({\mathcal {I}} \leqL\) which is acceptable. Hence we may assume that k≥1, and thus d≥2.
We now perform a technical reduction to eliminate some distinctions between the real and complex forms of the variety ℓ. We may apply Proposition 4.6 to each variety ℓ. Those varieties ℓ which obey conclusion (i) of that proposition can be easily handled by the induction hypothesis, since the varieties W arising from that conclusion have dimension strictly less than k. Thus we may restrict attention to varieties which obey conclusion (ii), namely that for every smooth real point p in ℓ (and in particular, for all \((p,\ell) \in{\mathcal{I}}\)), the real tangent space T _{ p } ℓ has full dimension k.
We now divide into two subcases: the case d>2k and the case d=2k, and deduce each case from the induction hypothesis. The main case is the latter; the former case will be obtainable from the induction hypothesis by a standard projection argument.
5.1 The Case of Excessively Large Ambient Dimension
We first deal with the case d>2k. Here, we can use a generic projection argument to deduce the theorem from the induction hypothesis. Indeed, fix ε,C _{0},P,L, and let π:R ^{ d }→R ^{2k } be a generic linear transformation (avoiding a finite number of positive codimension subvarieties of the space \(\operatorname {Hom}({\mathbf{R}}^{d},{\mathbf{R}}^{2k}) \equiv{\mathbf{R}}^{2kd}\) of linear transformations from R ^{ d } to R ^{2k }). Since k≥1, we see that for any two distinct points p,p′∈P, the set of transformations π for which π(p)=π(p′) has positive codimension in \(\operatorname{Hom}({\mathbf {R}}^{d},{\mathbf{R}}^{2k})\). Hence for generic π, the map from P to π(P) is bijective.
Similarly, let ℓ,ℓ′ be two distinct kdimensional varieties in L. Then generically π(ℓ),π(ℓ′) will be kdimensional varieties in R ^{2k }. We claim that for generic π, the varieties π(ℓ),π(ℓ′) are distinct. Indeed, since ℓ,ℓ′ are distinct varieties of the same dimension, there exists a point p on ℓ′ that does not lie on ℓ. Since ℓ has codimension dk>d2k, we conclude that for generic π, the d2kdimensional plane π ^{1}(π(p)) does not intersect ℓ′, and the claim follows. Thus the map π:L→π(L) is bijective.
Next, let \((p,\ell) \in{\mathcal{I}}\). Clearly, π(p)∈π(ℓ), and so π induces a bijection from \({\mathcal{I}}\) to some set of incidences \(\pi ({\mathcal{I}})\subset I(\pi(P), \pi(L))\).
If ℓ∈L, then ℓ is a kdimensional variety of degree at most C _{0}, and so generically π(ℓ) will also be a kdimensional variety of degree at most C _{0}, which gives Axiom (i).
Axioms (ii) and (iii) for \(\pi(P), \pi(L), \pi({\mathcal{I}})\) are generically inherited from those of \(P, L, {\mathcal{I}}\) thanks to the bijection between \({\mathcal{I}}\) and \(\pi({\mathcal{I}})\).
Now we turn to Axiom (iv). Fix \((p,\ell) \in{\mathcal{I}}\), then p is a smooth point of ℓ. Since d>2k and k≥1, it will generically hold that the codimension 2k affine space π ^{1}(π(p)) will intersect the codimension dk variety ℓ only at p. Thus, π(p) will generically be a smooth point of π(ℓ), which gives Axiom (iv).
Finally, if \((p,\ell), (p,\ell') \in{\mathcal{I}}\), then by hypothesis, the kdimensional tangent spaces T _{ p } ℓ,T _{ p } ℓ′ are transverse and thus span a 2kdimensional affine space through p. Generically, π will be bijective from this space to ℂ^{2k }, and thus T _{ π(p)} π(ℓ)=π(T _{ p } ℓ) and T _{ π(p)} π′(ℓ′)=π(T _{ p } ℓ′) remain transverse. This gives Axiom (v).
5.2 The Case of Sharp Ambient Dimension

C _{1} is assumed to be sufficiently large depending on C _{0}, k and ε;

C _{2} is assumed to be sufficiently large depending on C _{1},C _{0},k, and ε; and

C _{3} is assumed to be sufficiently large depending on C _{2},C _{1},C _{0},k, and ε.
Suppose \(P, L, {\mathcal{I}}\) obey all the specified axioms. We begin with two standard trivial bounds:
Lemma 5.1
(Trivial Bounds)
Proof
One can also view this lemma as a special case of the classical Kővári–Sós–Turán theorem [26].
Following Guth and Katz [18], the next step is to apply the polynomial ham sandwich theorem of Stone and Tukey [45]. We will sketch their argument here. For more details we refer to the excellent review of the polynomial decomposition method in [25].
Theorem 5.2
(Polynomial Ham Sandwich Theorem)
As observed in [18], we may iterate this to obtain
Corollary 5.3
(Cell Decomposition [18])
Proof
Let ℓ be a variety in L, then ℓ is kdimensional and has degree O(1). Meanwhile, the set {Q=0} is a hypersurface of degree at most C _{1}. We conclude that ℓ either lies in {Q=0}, or intersects {Q=0} in an algebraic set of dimension at most k1. In the former case, ℓ cannot belong to any of the L _{ i }. In the latter case we apply a generalization of a classical result established independently by Oleinik and Petrovsky [33], Milnor [30], and Thom [51], such that the number of connected components of ℓ\{Q=0} is at most \(O( C_{1}^{k} )\); we give a proof of this fact in Theorem A.2. Recently a more general bound was proved by Barone and Basu [4].
From Theorem A.2 we will use here that ℓ can belong to at most \(O(C_{1}^{k})\) of the sets L _{ i }.
Remark 5.4
A modification of the above argument shows that if one applied Corollary 5.3 not with a bounded degree D=C _{1}, but instead with a degree D comparable to (P^{2/3}L^{1/3})^{1/k }, then one could control the contribution of the cell interiors Ω _{ i } by the trivial bound (5.2), rather than the inductive hypothesis, thus removing the need to concede an epsilon in the exponents. However, the price one pays for this is that the hypersurface {Q=0} acquires a much higher degree, and the simple arguments given below to handle the incidences on this hypersurface are insufficient to give good bounds (except in the original Szemerédi–Trotter context when k=1 and d=2). Nevertheless, it may well be that a more careful analysis, using efficient quantitative bounds on the geometry of highdegree varieties, may be able to recover good bounds for this strategy, and in particular in removing the epsilon loss in Theorem 2.1.
For inductive reasons, it will be convenient to prove the following generalisation:
Proposition 5.5
Clearly, (5.4) follows by specialising to the case r=2k1, D=C _{1}, and Σ={Q=0}.
Proof
We induct on r. If r=0, then Σ is a single point, and so each set ℓ in L has at most one incidence in P∩Σ, giving the net bound \({\mathcal{I}} \leqL\), which is acceptable.
Now suppose that 1≤r<2k1, and that the claim has already been proven for smaller values of r.
By deleting the singular points of Σ from P, we may thus assume without loss of generality that all the points in P are smooth points of Σ. For each ℓ∈L, consider the intersection ℓ∩Σ. As ℓ is a variety of degree k, we see that ℓ is either contained in Σ, or ℓ∩Σ will be a algebraic set of dimension strictly less than k.
Consider the contribution of the first case when ℓ is contained in Σ. If we have two distinct incidences \((p,\ell), (p,\ell')\in{\mathcal{I}}\) such that ℓ,ℓ′ both lie in Σ, then the kdimensional tangent spaces T _{ p } ℓ,T _{ p } ℓ′ lie in the rdimensional space T _{ p } Σ. On the other hand, by Axiom (iv), T _{ p } ℓ and T _{ p } ℓ′ are transverse. Since r<2k, this is a contradiction. Thus, each point in P is incident to at most one variety ℓ∈L that lies in Σ, and so there are at most L incidences that come from this case.
We may thus assume that each variety ℓ∈L intersects Σ in an algebraic set of dimension strictly less than k. Since ℓ has degree at most C _{0}, and Σ has degree at most D, we see from Corollary 4.5 that ℓ∩Σ is the union of the smooth points of O _{ D }(1) algebraic varieties of dimension between 0 and k1 and degree O _{ D }(1).
Fix k′ and j. Those varieties ℓ∈L for which \({\mathcal {I}}\cap(P\times\{\ell\}) \leq C_{0}\) will contribute at most C _{0}L incidences to (5.5), so we may assume that \({\mathcal{I}}\cap(P \times \{\ell\}) >C_{0}\) for all ℓ∈L. By Axiom (ii), this forces the ℓ _{ k′,j } to be distinct. If we then let \(L' = L'_{k',j}\) be the set of all the ℓ _{ k′,j }, we can thus identify \({\mathcal{I}}_{k',j}\) with a subset \({\mathcal{I}}'\) of I(P,L′). However, by induction hypothesis, Theorem 2.1 is already known to hold if k is replaced by k′ (keeping d=2k fixed). So, to conclude the argument, it suffices to show that \(P, L', {\mathcal {I}}'\) obey the axioms of Theorem 2.1, with C _{0} replaced by \(O_{C_{0},D}(1)\). But Axiom (i) is clear from construction, while Axioms (ii), (iii), (iv), and (v) are inherited from the corresponding axioms for \(P, L,{\mathcal{I}}\). This closes the induction and proves the lemma. □
The proof of Theorem 2.1 is now complete.
Remark 5.6
It is reasonable to conjecture that one can set ε=0 in Theorem 2.1. From k=1, this can be established using the crossing number inequality [2, 28] and the Harnack curve theorem [19], following the arguments of Székely [46]; it is also possible to establish this inequality via the polynomial partitioning method. For k=2, a more careful analysis of the above arguments (using the refinement to the k=1 case mentioned above) eventually shows that one can take A to be of the shape \(\exp(O_{C_{0}}(1/\varepsilon))\); optimizing in ε, one can thus replace the AP^{ ε } factor by \(\exp( O_{C_{0}}( \sqrt{\logP} ) )\). However, for k>2, the highly inductive nature of the argument causes A to depend on ε in an iterated exponential manner.
Remark 5.7
One can construct many examples in which I(P,L) is comparable to \(P^{\frac{2}{3}} L^{\frac{2}{3}} + P + L\) by taking k of the standard pointline configurations in R ^{2} that demonstrate that the original Szemerédi–Trotter theorem (Theorem 1.1) is sharp, and then taking Cartesian products (and increasing the ambient dimension if desired). It is natural to conjecture that the ε loss in (2.1) can be eliminated, but our methods do not seem to easily give this improvement.
It is possible to drop Axiom (iv), at the cost of making Axiom (v) more complicated. For any point p on a real algebraic variety ℓ⊂R ^{ d }, define the tangent cone C _{ p } ℓ to be the set of all elements in R ^{ d } of the form γ′(0), where γ:[1,1]→ℓ is a smooth map with γ(0)=p. Note that at a smooth point p of ℓ, the tangent cone is nothing more than the tangent space T _{ p } ℓ (translated to the origin). However, the tangent cone continues to be welldefined at singular points, while the tangent space is not.
Corollary 5.8
The conclusions of Theorem 2.1 continue to hold if Axiom (iv) is dropped, but the tangent space T _{ p } ℓ in Axiom (v) is replaced by the tangent cone, but where the constant A is now allowed to depend on the ambient dimension d in addition to k and ε.
Proof
(Sketch) We perform strong induction on k, assuming that the claim has already been proven for smaller k. For those incidences (p,ℓ) in \({\mathcal{I}}\) for which p is a smooth point of ℓ, one can apply Theorem 2.1 to get a good bound, so we may restrict attention to those incidences in which p is a singular point of ℓ. But then we can use Corollary 4.5 to cover the singular portion of ℓ by \(O_{C_{0},k,d}(1)\) irreducible components in ℓ of dimension at most k1 and degree \(O_{C_{0},k,d}(1)\). Applying the induction hypothesis to each of these components (noting that Axioms (i)–(iii) and the modified Axiom (v) are inherited by these components, increasing C _{0} if necessary) we obtain the claim. □
Footnotes
 1.
It is possible for these real tangent spaces to have dimension less than k, because they are the restriction of the kdimensional complex tangent spaces T _{ p } ℓ _{ℂ}, \(T_{p}\ell'_{\mathbb{C}}\) to R ^{ d }. In applications, the complex tangent spaces will be complexifications of the real tangent spaces, which are then necessarily kdimensional; see Proposition 4.6.
 2.
Here we use the usual notation: C+D={a+ba∈C,b∈D} and CD={aba∈C,b∈D}.
 3.
Strictly speaking, because two different complex varieties may have the same real points, a real algebraic variety should really be viewed as a pair (V _{ R },V) rather than just the set V _{ R }; however, we will abuse notation and identify a real algebraic variety with the set V _{ R } of real points.
 4.
Here, we use the term “complexity” informally to refer to the number and degree of polynomials needed to define the variety.
 5.
One can also use the real inverse function theorem here, by viewing P _{1} as a map from R ^{2d } to R ^{2} instead of from ℂ^{ d } to ℂ, and noting that the hypothesis ∇P _{1}(x)≠0 ensures that the real derivative of P _{1} (which is a 2d×2 real matrix) has full rank.
 6.
Notes
Acknowledgements
The authors are very grateful to Boris Bukh, Nets Katz, Jordan Ellenberg, and Josh Zahl for helpful discussions and to the Isaac Newton Institute, Cambridge for hospitality while this research was being conducted. We also thank Alex Iosevich, Izabella Łaba, Jiří Matoušek, and János Pach for comments, references, and corrections to an earlier draft of this manuscript. We are thankful for the referee for the careful reading and for the suggestions to improve the readability of the paper. The first author is supported by an NSERC grant and the second author is supported by a grant from the MacArthur Foundation, by NSF grant DMS0649473, and by the NSF Waterman award.
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