Constructions for the Elekes–Szabó and Elekes–Rónyai problems

We give a construction of a non-degenerate polynomial F ∈ R[x, y, z] and a set A of cardinality n such that |Z(F ) ∩ (A×A×A)| n 3 2 , thus providing a new lower bound construction for the Elekes–Szabó problem. We also give a related construction for the Elekes–Rónyai problem restricted to a subgraph. This consists of a polynomial f ∈ R[x, y] that is not additive or multiplicative, a set A of size n, and a subset P ⊂ A×A of size |P | n3/2 on which f takes only n distinct values. Mathematics Subject Classifications: 52C10


Introduction
Throughout this paper, we write X Y if and only if there exists some absolute constant c > 0 such that X cY . If the constant c depends on another parameter k, we use the shorthand X k Y . Given a polynomial F (x 1 , . . . , x n ) ∈ R[x 1 , . . . , x n ], Z(F ) = {(x 1 , . . . , x n ) : F (x 1 , . . . , x n ) = 0} denotes the zero set of F .

The Elekes-Szabó Problem
Elekes and Szabó [6] considered the size of the intersection of the zero set of a polynomial F (x, y, z) ∈ R[x, y, z] of degree d with a Cartesian product A × B × C ⊂ R 3 , where |A| = |B| = |C| = n. By the Schwartz-Zippel Lemma (see for instance [9,Lemma A.4 This bound cannot be improved in general. For example, if F (x, y) = x + y + z, A = B = {1, . . . , n}, and C = {−1, . . . , −n}, then |Z(F ) ∩ (A × B × C)| n 2 . More generally, if the equation F (x, y, z) = 0 is in some sense equivalent to an equation of the form ϕ 1 (x) + ϕ 2 (y) + ϕ 3 (z) = 0, then we can choose A, B, C so that |Z(F ) ∩ (A × B × C)| n 2 . The following definition makes this property precise.
is degenerate if there are intervals I 1 , I 2 , I 3 , and for each i there is a smooth (infinitely differentiable) function ϕ i : I i → R which has a smooth inverse, such that for all (x, y, z) ∈ I 1 × I 2 × I 3 we have F (x, y, z) = 0 if and only if ϕ 1 (x) + ϕ 2 (y) + ϕ 3 (z) = 0.
Elekes and Szabó [6] showed that if the polynomial is not degenerate in this sense, then the bound (1) can be improved to n 2−η for some η > 0. A quantitative improvement to η = 1/6 was obtained by Raz, Sharir and de Zeeuw [9], leading to the following statement.
Theorem 2 ( [6,9]). Let F ∈ R[x, y, z] be a polynomial of degree d. If F is not degenerate, then for any A, B, C ⊂ R of size n we have Not much attention has been paid to lower bound constructions for this theorem. Elekes [3] noted that for F = x 2 + xy + y 2 − z and A = {1, . . . , n} we have |Z(F ) ∩ (A × A × A)| n √ log n (actually, Elekes formulated this in a different way, which we mention in the next section; see [15] for more discussion). This was the only known lower bound for Theorem 2, and some have suggested that the upper bound could be improved as far as O(n 1+ε ) for an arbitrarily small ε > 0; for instance, the fourth author wrote this in [15].
The main purpose of this paper is to show by means of a simple example that this is not the case, and that in fact the bound in Theorem 2 cannot be improved beyond O(n 3/2 ). Our main result is the following theorem.
Theorem 3. There exists a polynomial F ∈ R[x, y, z] of degree 2 that is not degenerate, such that for any n there is a set A ⊂ R of size n with In Section 4, we briefly discuss possible extensions of this theorem to polynomials in more variables.
The construction of the polynomial F in the above statement is closely related to a construction of Valtr [14], which first appeared in the context of the Erdős unit distance problem. Other constructions throughout this paper also use similar ideas.

The Elekes-Rónyai Problem
Before the work of Elekes and Szabó [6], Elekes and Rónyai [5] considered the question of bounding the image of a polynomial f ∈ R[x, y] restricted to a Cartesian product, assuming that f does not have a certain special form, which is specified in the following definition.
Elekes and Rónyai [5] proved that if f ∈ R[x, y] is not additive or multiplicative, then for every A, B ⊆ R with |A| = |B| = n the image |f (A, B)| is superlinear in n. The current state of the art for this problem is the following result of Raz, Sharir and Solymosi [8].
Elekes [3] noted that if f (x, y) = x 2 + xy + y 2 and A = {1, . . . , n}, then |f (A, A)| n 2 / √ log n. This is the best known upper bound construction for Theorem 5, which suggests that we may have |f (A, B)| n 2− for all positive . This conjecture is widely believed, see for instance Elekes [3] or Matoušek [7,Section 4.1]. The construction that we give in the proof of Theorem 3 does not translate into a construction that disproves this conjecture.
Nevertheless, we show that there is a polynomial that takes only a linear number of values on a certain large subset of the pairs in A × A. This approach is partly inspired by work of Alon, Ruzsa and Solymosi [1] concerning constructions for the sum-product problem along graphs. See also [12] for a slightly improved construction.
Let G be a bipartite graph on A and B with edge set Our result is the following. Theorem 6. There exists a polynomial f ∈ R[x, y] of degree 2 that is not additive or multiplicative, a finite set A ⊂ R of size n, and a bipartite graph G on A × A, such that |E(G)| n 3/2 and |f G (A, B)| n.

The Elekes-Szabó problem
In this section we prove Theorem 3. Define the electronic journal of combinatorics 27(1) (2020), #P1.57 We set A = {1, . . . , n} and we consider the intersection of F with A × A × A. Consider the subset Each choice of k and determines a distinct triple in T , and so we have |T | n 3/2 . For each triple in T , we have It remains to show that F is not degenerate in the sense of Definition 1. We will use an idea introduced by Elekes and Rónyai [5], which is that this type of degeneracy can be verified using the following straightforward derivative test; see for instance [6,Lemma 33]  Lemma 7. Let f : R 2 → R be a smooth function on some open set U ⊂ R 2 with f x and f y not identically zero. If there exist smooth functions ψ, ϕ 1 , ϕ 2 on U such that is identically zero on U .
We now check if the expression (2) is identically zero on I 1 × I 2 × I 3 . We have This expression equals zero only when y − x = 1/4, so it does not vanish on any nontrivial open set. Thus (2) is not identically zero, and by Lemma 7 this contradicts our assumption that F is degenerate.

The Elekes-Rónyai problem along a graph
We now prove Theorem 6, concerning the image of a polynomial along a subset of a Cartesian product.
Set A = {1, . . . , n} and let G be the bipartite graph on A × A with the edge set We have |E(G)| n 3/2 . Applying f along any edge gives a non-negative integer This shows that |f G (A × A)| n.
It remains to prove that f is not additive or multiplicative. We could again do this using Lemma 7, but here we can use a more elementary approach. We treat the two cases separately.
Additive case: Suppose f (x, y) = g(h(x) + k(y)). Note that g, h and k must have degree at most 2. We cannot have deg(g) = 1, since then f (x, y) would not have any cross term xy. If deg(g) = 2, then deg(h) = deg(k) = 1. We can write with b 1 and c 1 non-zero. Then we have Calculating the coefficient for the y term on the right hand side and comparing with the left hand side, it follows that On the other hand, calculating the coefficient for the x term on the right hand side of (3) and comparing with the left hand side, it follows that b 1 (2a 2 (b 0 + c 0 ) + a 1 ) = 1.
Multiplicative case: Suppose f (x, y) = g(h(x) · k(y)). We cannot have deg(g) = 2, since then h or k would have to be constant, and f (x, y) would not depend on both variables. Therefore we have deg(g) = 1. In this case, we must have deg(h) = deg(k) = 1. We can write This is a contradiction, since there is no x 2 or y 2 term on the right hand side. This completes our proof that f is not additive or multiplicative, which completes our proof of Theorem 5.

.1 Four variables
One can consider the same problems for polynomials in more variables. Raz, Sharir and de Zeeuw [10] proved that for F ∈ R[x, y, s, t] of degree d and A, B, C, D ⊂ R of size n, we have unless F (x, y, s, t) = 0 is in a local sense (similar to Definition 1) equivalent to an equation of the form ϕ 1 (x) + ϕ 2 (y) + ϕ 3 (s) + ϕ 4 (t) = 0. A construction of Valtr [14] (see also [13,Section 5.3]) essentially shows that for It is not hard to verify (as in our proof of Theorem 3) that V (x, y, s, t) is not degenerate in the sense of [10], so this gives a lower bound construction for (5), which is the best known. Note that the polynomial F (x, y, z) in our proof of Theorem 3 can be obtained from Valtr's polynomial V (x, y, s, t) by setting s = x and t = z.

More than four variables
For more than four variables, we do not have a statement that is entirely analogous to Theorem 2 or (5). Bays and Breuillard [2] proved a similar statement for any number of variables, but without an explicit exponent, and with a different description of the exceptional form. Also, Raz and Tov [11] extended Theorem 5 to any number of variables, with an explicit exponent.
Because for the Elekes-Szabó problem in more than four variables we do not have explicit exponents, and also because the appropriate definition of degeneracy is not clear, we only briefly touch on constructions for more variables here.
Then we have T ⊂ Z(F ) ∩ A m , which implies This should be compared with the Schwartz-Zippel bound |Z(F ) ∩ A m | n m−1 . A potential Elekes-Szabó theorem in m variables, i.e. an explicit version of the result of Bays and Breuillard, would give a bound of the form |Z(F ) ∩ A m | n m−1−ηm for some η m > 0, under the condition that F is not degenerate in some sense. Presuming that our polynomial F is not of this form, it would show that we must have η m 1/2.